Skip to main content

Full text of "An introduction to mathematics, by A. N. Whitehead"

See other formats


a .wo.H 

' 



OME UNIVERSITY LIBRARY 
F MODERN KNOWLEDGE 



AN INTRODUCTION TO 
MATHEMATICS 

BY A. N. WHITEHEAD, Se.D., F.R.S. 



LONDON 
WILLIAMS & NORGATE 



HKNRY HOLT & Co., NEW YORK 
CANADA: WM. BRIGGS, TORONTO 
INDIA : R. & T. WASHBOURNE, LTD. 




HOME 

UNIVERSITY 

LIBRARY 

OF 

MODERN KNOWLEDGE 



Editor* t 

HERBERT FISHER, M.A.. F.B.A. 
PROF. GILBERT MURRAY, D.LlTT., 

LL.D., F.B.A. 

PROF. J. ARTHUR THOMSON, M.A. 
PROF. WILLIAM T. BKEWSTER, M.A. 

COLUMBIA. UNIVERSITY, U.S.A.) 











NEW YORK 

HENRY HOLT AND COMPANY 






, 






iifr A /uilfJ 



I 



AN 



INTRODUCTION 



TO 



MATHEMATICS 



BY 



A. N. WHITEHEAD, 
SC.D- F.R.S., 






AUTHOR OF "UNIVERSAL ALGEBRA," JOINT 
AUTHOR OF " I'RINCIPIA MATHBHATICA " 



NEW AND REVISED EDITION 



IRV^T 







1 


LONDON 

WILLIAMS AND NORGATE 









PRINTED BT 

HAKELJy, WATSON AND VINET, 
LONDON AND AYLBSBUBT. 



CONTENTS 

OH&P. PASH 

I THE ABSTRACT NATURE OF MATHE- 
MATICS ..... 7 

H VARIABLES . . , . . 15 

m METHODS OF APPLICATION . . 25 

IV DYNAMICS ..... 42 

V THE SYMBOLISM OF MATHEMATICS . 58 

VI GENERALIZATIONS OF NUMBER . . 71 

VH IMAGINARY NUMBERS ... 87 

VIH IMAGINARY NUMBERS (CONTINUED) . 101 

IX COORDINATE GEOMETRY . . .112 

X CONIC SECTIONS . . . .128 

XI FUNCTIONS . . . . . 145 

XII PERIODICITY IN NATURE . . .164 

v 



vi CONTENTS 

CHAP. PAGE 

TRIGONOMETRY . . . .173 



XIV SERIES .,..., 194 

XV THE DIFFERENTIAL CALCULUS . . 217 

XVI GEOMETRY . . . . 236 

XVH QUANTITY . . . . . 245 

NOTES ...... 250 

BIBLIOGRAPHY . . .251 

INDEX 253 



AN INTEODUCTION TO 
MATHEMATICS 

CHAPTER I 

THE ABSTRACT NATURE OF MATHEMATICS 

THE study of mathematics is apt to com' 
mence in disappointment. The important 
applications of the science, the theoretical 
interest of its ideas, and the logical rigour of 
its methods, all generate the expectation of 
a speedy introduction to processes of interest. 
We are told that by its aid the stars are 
weighed and the billions of molecules in a 
drop of water are counted. Yet, like the 
ghost of Hamlet's father, this great science 
eludes the efforts of our mental weapons 
to grasp it " 'Tis here, 'tis there, 'tis 
gone " and what we do see does not suggest 
the same excuse for illusiveness as sufficed 
for the ghost, that it is too noble for 
our gross methods. '* A show of violence," 
if ever excusable, may surely be " offered " 
to the trivial results which occupy the 
7 



8 INTRODUCTION TO MATHEMATICS 

pages of some elementary mathematical 
treatises. 

The reason for this failure of the science to 
live up to its reputation is that its funda- 
mental ideas are not explained to the student 
disentangled from the technical procedure 
which has been invented to facilitate their 
exact presentation in particular instances. 
Accordingly, the unfortunate learner finds 
himself struggling to acquire a knowledge of 
a mass of details which are not illuminated 
by any general conception. Without a doubt, 
technical facility is a first requisite for valu- 
able mental activity : we shall fail to appre- 
ciate the rhythm of Milton, or the passion of 
Shelley, so long as we find it necessary to 
spell the words and are not quite certain of 
the forms of the individual letters. In this 
sense there is no royal road to learning. But 
it is equally an error to confine attention to 
technical processes, excluding consideration 
of general ideas. Here lies the road to 
pedantry. 

The object of the following Chapters is not 
to teach mathematics, but to enable students 
from the very beginning of their course to 
know what the science is about, and why it is 
necessarily the foundation of exact thought 
as applied to natural phenomena. All allu- 
sion in what follows to detailed deductions 
in any part of the science will be inserted 



NATURE OF MATHEMATICS 9 

merely for the purpose of example, and care 
will be taken to make the general argument 
comprehensible, even if here and there some 
technical process or symbol which the reader 
does not understand is cited for the purpose 
of illustration. 

The first acquaintance which most people 
have with mathematics is through arithmetic. 
That two and two make four is usually taken 
as the type of a simple mathematical pro- 
position which everyone will have heard of. 
Arithmetic, therefore, will be a good subject 
to consider in order to discover, if possible, 
the most obvious characteristic of the science. 
Now, the first noticeable fact about arithmetic 
is that it applies to everything, to tastes and 
to sounds, to apples and to angels, to the 
ideas of the mind and to the bones of the 
body. The nature of the things is perfectly 
indifferent, of all things it is true that two 
and two make four. Thus we write down as 
the leading characteristic of mathematics 
that it deals with properties and ideas 
which are applicable to things just because 
they are things, and apart from any particular 
feelings, or emotions, or sensations, in any 
way connected with them. This is what 
is meant by calling mathematics an abstract 
science. 

The result which we have reached deserves 
attention. It is natural to think that an 



10 INTRODUCTION TO MATHEMATICS 

abstract science cannot be of much import- 
ance in the affairs of human life, because it 
has omitted from its consideration every- 
thing of real interest. It will be remembered 
that Swift, in his description of Gulliver's 
voyage to Laputa, is of two minds on this 
point. He describes the mathematicians of 
that country as silly and useless dreamers, 
whose attention has to be awakened by 
flappers. Also, the mathematical tailor mea- 
sures his height by a quadrant, and deduces 
his other dimensions by a rule and compasses, 
producing a suit of very ill-fitting clothes. 
On the other hand, the mathematicians of 
Laputa, by their marvellous invention of the 
magnetic island floating in the air, ruled the 
country and maintained their ascendency 
over their subjects. Swift, indeed, lived at 
a time peculiarly unsuited for gibes at con- 
temporary mathematicians. Newton's Prin- 
cipia had just been written, one of the great 
forces which have transformed the modern 
world. Swift might just as well have laughed 
at an earthquake. 

But a mere list of the achievements of 
mathematics is an unsatisfactory way of 
arriving at an idea of its importance. It is 
worth while to spend a little thought in 
getting at the root reason why mathematics, 
because of its very abstractness, must always 
remain one of the most important topics 



NATURE OF MATHEMATICS 11 

for thought. Let us try to make clear to 
ourselves why explanations of the order of 
events necessarily tend to become mathe- 
matical. 

Consider how all events are interconnected. 
When we see the lightning, we listen for the 
thunder ; when we hear the wind, we look 
for the waves on the sea ; in the chill autumn, 
the leaves fall. Everywhere order reigns, so 
that when some circumstances have been 
noted we can foresee that others will also be 
present. The progress of science consists in 
observing these interconnections and in show- 
ing with a patient ingenuity that the events 
of this evershifting world are but examples of 
a few general connections or relations called 
laws. To see what is general in what is par- 
ticular and what is permanent in what is 
transitory is the aim of scientific thought. In 
the eye of science, the fall of an apple, the 
motion of a planet round a sun, and the cling- 
ing of the atmosphere to the earth are all 
seen as examples of the law of gravity. This 
possibility of disentangling the most complex 
evanescent circumstances into various ex- 
amples of permanent laws is the controlling 
idea of modern thought. 

Now let us think of the sort of laws which 
we want in order completely to realize this 
scientific ideal. Our knowledge of the par- 
ticular facts of the world around us is gained 



12 INTRODUCTION TO MATHEMATICS 

from our sensations. We see, and hear, and 
taste, and smell, and feel hot and cold, and 
push, and rub, and ache, and tingle. These 
are just our own personal sensations : my 
toothache cannot be your toothache, and my 
sight cannot be your sight. But we ascribe 
the origin of these sensations to relations be- 
tween the things which form the external 
world. Thus the dentist extracts not the 
toothache but the tooth. And not only so, 
we also endeavour to imagine the world as 
one connected set of things which underlies 
all the perceptions of all people. There is not 
one world of things for my sensations and an- 
other for yours, but one world in which we 
both exist. It is the same tooth both for 
dentist and patient. Also we hear and we 
touch the same world as we see. 

It is easy, therefore, to understand that we 
want to describe the connections between 
these external things in some way which does 
not depend on any particular sensations, nor 
even on all the sensations of any particular 
person. The laws satisfied by the course of 
events in the world of external things are to 
be described, if possible, in a neutral uni- 
versal fashion, the same for blind men as for 
deaf men, and the same for beings with 
faculties beyond our ken as for normal human 
beings. 

But when we have put aside our immediate 



NATURE OF MATHEMATICS 13 

sensations, the most serviceable part from 
its clearness, definiteness, and universality 
of what is left is composed of our general ideas 
of the abstract formal properties of things; 
in fact, the abstract mathematical ideas men- 
tioned above. Thus it comes about that, 
step by step, and not realizing the full mean- 
ing of the process, mankind has been led to 
search for a mathematical description of the 
properties of the universe, because in this way 
only can a general idea of the course of events 
be formed, freed from reference to particular 
persons or to particular types of sensation. 
For example, it might be asked at dinner: 
" What was it which underlay my sensation 
of sight, yours of touch, and his of taste 
and smell ? " the answer being " an apple." 
But in its final analysis, science seeks to 
describe an apple in terms of the positions 
and motions of molecules, a description which 
ignores me and you and him, and also ig- 
nores sight and touch and taste and smell. 
Thus mathematical ideas, because they 
are abstract, supply just what is wanted 
jfor a scientific description of the course 01 
events. 

This point has usually been misunderstood, 
from being thought of in too narrow a way. 
Pythagoras had a glimpse of it when he pro- 
claimed that number was the source of all 
things. In modern times the belief that the 



14 INTRODUCTION TO MATHEMATICS 

ultimate explanation of all things was to be 
found in Newtonian mechanics was an adum- 
bration of the truth that all science as it 
grows towards perfection becomes mathe- 
matical in its ideas. 



CHAPTER II 

VARIABLES 

MATHEMATICS as a science commenced when 
first someone, probably a Greek, proved pro- 
positions about any things or about some 
things, without specification of definite par- 
ticular things. These propositions were first 
enunciated by the Greeks for geometry ; and, 
accordingly, geometry was the great Greek 
mathematical science. After the rise of geo- 
metry centuries passed away before algebra 
made a really effective start, despite some 
faint anticipations by the later Greek mathe- 
maticians. 

The ideas of any and of some are intro- 
duced into algebra by the use of letters, in- 
stead of the definite numbers of arithmetic. 
Thus, instead of saying that 2+3=3+2, in 
algebra we generalize and say that, if x and 
y stand for any two numbers, then x -\-y =y +a?. 
Again, in the place of saying that 3 > 2, we 
generalize and say that if x be any number 
there exists some number (or numbers) y such 
that y>x. We may remark in passing that 
this latter assumption for when put in its 
strict ultimate form it is an assumption is 

15 



16 INTRODUCTION TO MATHEMATICS 

of vital importance, both to philosophy and 
to mathematics ; for by it the notion of in- 
finity is introduced. Perhaps it required the 
introduction of the arabic numerals, by which 
the use of letters as standing for definite 
numbers has been completely discarded in 
mathematics, in order to suggest to mathe- 
maticians the technical convenience of the 
use of letters for the ideas of any number 
and some number. The Romans would have 
stated the number of the year in which this 
is written in the form MDCCCCX., whereas 
we write it 1910, thus leaving the letters for 
the other usage. But this is merely a specu- 
lation. After the rise of algebra the differ- 
ential calculus was invented by Newton and 
Leibniz, and then a pause in the progress 
of the philosophy of mathematical thought 
occurred so far as these notions are concerned ; 
and it was not till within the last few years 
that it has been realized how fundamental 
any and some are to the very nature of mathe- 
matics, with the result of opening out still 
further subjects for mathematical explora- 
tion. 

Let us now make some simple algebraic 
statements, with the object of understanding 
exactly how these fundamental ideas occur. 

(1) For any number a?, #+2=2+#; 

(2) For some number #, #+2=8 ; 

(3) For some number a?, x +2 > 3. 



VARIABLES 17 

The first point to notice is the possibilities 
contained in the meaning of some, as here 
used. Since a +2 =2 +x for any number a?, it 
is true for some number x. Thus, as here used, 
any implies some and some does not exclude 
any. Again, in the second example, there is, 
in fact, only one number x, such that x +2 =8, 
namely only the number 1. Thus the some 
may be one number only. But in the third, 
example, any number x which is greater than 
1 gives x -f 2 > 3. Hence there are an infinite 
number of numbers which answer to the some 
number in this case. Thus some may be any- 
thing between any and one only, including 
both these limiting cases. 

It is natural to supersede the statements 
(2) and (3) by the questions : 

(2') For what number x is x +2 =3; 

(3') For what numbers x is #-f2>3. 
Considering (2'), #+2=3 is an equation, and 
it is easy to see that its solution is x =3 2 =1. 
When we have asked the question implied in 
the statement of the equation <r+2=8, x is 
called the unknown. The object of the solu- 
tion of the equation is the determination of 
the unknown. Equations are of great im- 
portance in mathematics, and it seems as 
though (2') exemplified a much more thorough- 
going and fundamental idea than the original 
statement (2). This, however, is a complete 
mistake. The idea of the undetermined 



18 INTRODUCTION TO MATHEMATICS 

" variable " as occurring in the use of ** some " 
or " any " is the really important one in 
mathematics ; that of the " unknown " in an 
equation, which is to be solved as quickly as 
possible, is only of subordinate use, though 
of course it is very important. One of the 
causes of the apparent triviality of much of 
elementary algebra is the preoccupation of 
the text-books with the solution of equations. 
The same remark applies to the solution of 
the inequality (3') as compared to the original 
statement (3). 

But the majority of interesting formulae, 
especially when the idea of some is present, 
involve more than one variable. For ex- 
ample, the consideration of the pairs of num- 
bers x and y (fractional or integral) which 
satisfy x+y=I involves the idea of two corre- 
lated variables, x and y. When two variables 
are present the same two main types of 
statement occur. For example, (1) for 
any pair of numbers, x and y, x+y=y+ac, 
and (2) for some pairs of numbers, x and t/, 



The second type of statement invites con- 
sideration of the aggregate of pairs of num- 
bers which are bound together by some fixed 
relation in the case given, by the relation 
x+y=\. One use of formulae of the first 
type, true for any pair of numbers, is that by 
them formulae of the second type can be 



VARIABLES 19 

thrown into an indefinite number of equiva- 
lent forms. For example, the relation x-\-y 
=1 is equivalent to the relations 

y+x=I, (x-y)+2y=l, 6x+6y=6, 

and so on. Thus a skilful mathematician 
uses that equivalent form of the relation 
under consideration which is most convenient 
for his immediate purpose. 

It is not in general true that, when a pair 
of terms satisfy some fixed relation, if one of 
the terms is given the other is also definitely 
determined. For example, when x and y 
satisfy y 2 =x, if #=4, y can be 2, thus, 
for any positive value of x there are alter- 
native values for y. Also in the relation 
j?+t/>l, when either x or y is given, an 
indefinite number of values remain open for 
the other. 

Again there is another important point to 
be noticed. If we restrict ourselves to posi- 
tive numbers, integral or fractional, in con- 
sidering the relation <c+t/=l, then, if either 
x or y be greater than 1, there is no positive 
number which the other can assume so as to 
satisfy the relation. Thus the "field" of 
the relation for x is restricted to numbers less 
than 1, and similarly for the " field " open 
to y. Again, consider integral numbers only, 
positive or negative, and take the relation 



20 INTRODUCTION TO MATHEMATICS 

t/ 2 =#, satisfied by pairs of such numbers. 
Then whatever integral value is given to y, 
x can assume one corresponding integral 
value. So the " field " for y is unrestricted 
among these positive or negative integers. 
But the " field " for x is restricted in two 
ways. In the first place x must be positive, 
and in the second place, since y is to be in- 
tegral, x must be a perfect square. Accord- 
ingly, the " field " of x is restricted to the set 
of integers I 2 , 2 2 , 3 2 , 4 2 , and so on, i.e., to 1, 
4, 9, 16, and so on. 

The study of the general properties of a 
relation between pairs of numbers is much 
facilitated by the use of a diagram constructed 
as follows : 




O x M i A 

Fig. 1. 



Draw two lines OX and OF at right angles ; 
let any number x be represented by x units 



VARIABLES 21 

(in any scale) of length along OX, any num- 
ber ybyy units (in any scale) of length along 
OF. Thus if OM, along OX, be x units in 
length, and ON, along OF, be y units in length, 
by completing the parallelogram OMPN we 
find a point P which corresponds to the pair 
of numbers x and y. To each point there 
corresponds one pair of numbers, and to each 
pair of numbers there corresponds one point. 
The pair of numbers are called the co- 
ordinates of the point. Then the points 
whose coordinates satisfy some fixed rela- 
tion can be indicated in a convenient way, 
by drawing a line, if they all lie on a line, 
or by shading an area if they are all points 
in the area. If the relation can be repre- 
sented by an equation such as o?+t/=l, or 
t/ 2 =#, then the points lie on a line, which is 
straight in the former case and curved in 
the latter. For example, considering only 
positive numbers, the points whose co- 
ordinates satisfy x-\-y=\ lie on the straight 
line AB in Fig. 1, where 0^=1 and OB=l. 
Thus this segment of the straight line AB 
gives a pictorial representation of the proper- 
ties of the relation under the restriction to 
positive numbers. 

Another example of a relation between two 
variables is afforded by considering the varia- 
tions in the pressure and volume of a given 
mass of some gaseous substance such as air 



22 INTRODUCTION TO MATHEMATICS 

or coal-gas or steam at a constant tempera- 
ture. Let v be the number of cubic feet in 
its volume and p its pressure in Ib. weight 
per square inch. Then the law, known as 
Boyle's law, expressing the relation between 
p and v as both vary, is that the product 
pv is constant, always supposing that the 
temperature does not alter. Let us suppose, 
for example, that the quantity of the gas 
and its other circumstances are such that 
we can put pv=I (the exact number on 
the right-hand side of the equation makes 
no essential difference). 




Then in Fig. 2 we take two lines, OV and 
OP, at right angles and draw OM along OV 
to represent v units of volume, and ON along 



VARIABLES 23 

OP to represent p units of pressure. Then 
the point Q, which is found by completing the 
parallelogram OMQN, represents the state of 
the gas when its volume is v cubic feet and its 
pressure is p Ib. weight per square inch. If 
the circumstances of the portion of gas con- 
sidered are such that pv=l, then all these 
points Q which correspond to any possible 
state of this portion of gas must lie on the 
curved line ABC, which includes all points 
for which p and v are positive, and pv=I. 
Thus this curved line gives a pictorial repre- 
sentation of the relation holding between the 
volume and the pressure. When the pressure 
is very big the corresponding point Q must 
be near C, or even beyond C on the undrawn 
part of the curve ; then the volume will be 
very small. When the volume is big Q will 
be near to A, or beyond A ; and then the 
pressure will be small. Notice that an en- 
gineer or a physicist may want to know the 
particular pressure corresponding to some 
definitely assigned volume. Then we have 
the case of determining the unknown p when 
v is a known number. But this is only in 
particular cases. In considering generally 
the properties of the gas and how it will be- 
have, he has to have in his mind the general 
form of the whole curve ABC and its general 
properties. In other words the really funda- 
mental idea is that of the pair of variables 



24 INTRODUCTION TO MATHEMATICS 

satisfying the relation pv=I. This example 
illustrates how the idea of variables is funda- 
mental, both in the applications as well as in 
the theory of mathematics. 



CHAPTER III 

METHODS OF APPLICATION 

THE way in which the idea of variables 
satisfying a relation occurs in the applications 
of mathematics is worth thought, and by 
devoting some time to it we shall clear up 
our thoughts on the whole subject. 

Let us start with the simplest of examples : 
Suppose that building costs Is. per cubic 
foot and that 205. make l. Then in all 
the complex circumstances which attend the 
building of a new house, amid all the various 
sensations and emotions of the owner, the 
architect, the builder, the workmen, and the 
onlookers as the house has grown to comple- 
tion, this fixed correlation is by the law 
assumed to hold between the cubic content 
and the cost to the owner, namely that if x 
be the number of cubic feet, and y the cost, 
then 20?/=a?. This correlation of x and y is 
assumed to be true for the building of any 
house by any owner. Also, the volume of 
the house and the cost are not supposed to 
have been perceived or apprehended by any 
particular sensation or faculty, or by any 

25 



26 INTRODUCTION TO MATHEMATICS 

particular man. They are stated in an ab- 
stract general way, with complete indiffer- 
ence to the owner's state of mind when he has 
to pay the bill. 

Now think a bit further as to what all this 
means. The building of a house is a com- 
plicated set of circumstances. It is im- 
possible to begin to apply the law, or to test 
it, unless amid the general course of events 
it is possible to recognize a definite set of 
occurrences as forming a particular instance 
of the building of a house. In short, we must 
know a house when we see it, and must recog- 
nize the events which belong to its building. 
Then amidst these events, thus isolated in 
idea from the rest of nature, the two elements 
of the cost and cubic content must be deter- 
minable ; and when they are both determined, 
if the law be true, they satisfy the general 
formula 

20y=a. 

*But is tl ! ( law true ? Anyone who has had 
much to i ; >. with building will know that we 
have hext put the cost rather high. It is 
only for <>a expensive type of house that it 
will work out at this price. This brings out 
another point which must be made clear. 
While we are making mathematical calcula- 
tions connected with the formula 20t/=#, it 
is indifferent to us whether the law be true or 



METHODS OF APPLICATION 27 

false. In fact, the very meanings assigned 
to x and y, as being a number of cubic feet 
and a number of pounds sterling, are in- 
different. During the mathematical investi- 
gation we are, in fact, merely considering the 
properties of this correlation between a pair 
of variable numbers x and y. OUT results 
will apply equally well, if we interpret y to 
mean a number of fishermen and x the num- 
ber of fish caught, so that the assumed law 
is that on the average each fisherman catches 
twenty fish. The mathematical certainty of 
the investigation only attaches to the results 
considered as giving properties of the corre- 
lation 20y=x between the variable pair of 
numbers x and y. There is no mathematical 
certainty whatever about the cost of the 
actual building of any house. The law is not 
quite true and the result it gives will not be 
quite accurate. In fact, it may well be hope- 
lessly wrong. 

Now all this no doubt seems very obvious. 
But in truth with more complicated instances 
there is no more common error than to assume 
that, because prolonged and accurate mathe- 
matical calculations have been made, the 
application of the result to some fact of 
nature is absolutely certain. The conclusion 
of no argument can be more certain than the 
assumptions from which it starts. All mathe- 
matical calculations about the course of 



28 INTRODUCTION TO MATHEMATICS 

nature must start from some assumed law of 
nature, such, for instance, as the assumed 
law of the cost of building stated above. 
Accordingly, however accurately we have 
calculated that some event must occur, the 
doubt always remains Is the law true ? If 
the law states a precise result, almost cer- 
tainly it is not precisely accurate ; and thus 
even at the best the result, precisely as calcu- 
lated, is not likely to occur. But then we 
have no faculty capable of observation with 
ideal precision, so, after all, our inaccurate 
laws may be good enough. 

We will now turn to an actual case, that 
of Newton and the Law of Gravity. This law 
states that any two bodies attract one an- 
other with a force proportional to the product 
of their masses, and inversely proportional to 
the square of the distance between them. 
Thus if m and M are the masses of the two 
bodies, reckoned in Ibs. say, and d miles is 
the distance between them, the force on either 
body, due to the attraction of the other and 

directed towards it, is proportional to p- ; 

thus this force can be written as equal to 

JT~ wnere & is a definite number depending 

on the absolute magnitude of this attraction 
and also on the scale by which we choose to 
measure forces. It is easy to see that, if we 



METHODS OF APPLICATION 29 

wish to reckon in terms of forces such as the 
weight of a mass of 1 lb., the number which 
k represents must be extremely small ; for 
when m and M and d are each put equal to 

1, becomes the gravitational attraction 

of two equal masses of 1 lb. at the distance of 
one mile, and this is quite inappreciable. 

However, we have now got our formula for 
the force of attraction. If we call this force 

F, it is F^kp-, giving the correlation be- 
tween the variables F, m, M, and d. We all 
know the story of how it was found out. 
Newton, it states^ was sitting in an orchard 
and watched the fall of an apple, and then 
the law of universal gravitation burst upon 
his mind. It may be that the final formu- 
lation of the law occurred to him in an 
orchard, as well as elsewhere and he must 
have been somewhere. But for our purposes 
it is more instructive to dwell upon the vast 
amount of preparatory thought, the product 
of many minds and many centuries, which 
was necessary before this exact law could be 
formulated. In the first place, the mathe- 
matical habit of mind and the mathematicaJ 
procedure explained in the previous two 
chapters had to be generated ; otherwise 
Newton could never have thought of a formula 
representing the force between any two masses 



30 INTRODUCTION TO MATHEMATICS 

at any distance. Again, what are the mean- 
ings of the terms employed, Force, Mass, Dis- 
tance ? Take the easiest of these terms, 
Distance. It seems very obvious to us to 
conceive all material things as forming a de- 
finite geometrical whole, such that the dis- 
tances of the various parts are measurable in 
terms of some unit length, such as a mile or 
a yard. This is almost the first aspect of a 
material structure which occurs to us. It is 
the gradual outcome of the study of geometry 
and of the theory of measurement. Even 
now, in certain cases, other modes of thought 
are convenient. In a mountainous country 
distances are often reckoned in hours. But 
leaving distance, the other terms, Force and 
Mass, are much more obscure. The exact 
comprehension of the ideas which Newton 
meant to convey by these words was of slow 
growth, and, indeed, Newton himself was the 
first man who had thoroughly mastered the 
true general principles of Dynamics. 

Throughout the middle ages, under the in- 
fluence of Aristotle, the science was entirely 
misconceived. Newton had the advantage of 
coming after a series of great men, notably 
Galileo, in Italy, who in the previous two 
centuries had reconstructed the science and 
had invented the right way of thinking about 
it. He completed their work. Then, finally, 
having the ideas of force, mass, and distance, 



clear and distinct in his mind, and realising 
their importance and their relevance to the 
fall of an apple and the motions of the planets, 
he hit upon the law of gravitation and proved 
it to be the formula always satisfied in these 
various motions. 

The vital point in the application of mathe- 
matical formulae is to have clear ideas and a 
correct estimate of their relevance to the 
phenomena under observation. No less than 
ourselves, our remote ancestors were im- 
pressed with the importance of natural 
phenomena and with the desirability of taking 
energetic measures to regulate the sequence 
of events. Under the influence of irrelevant 
ideas they executed elaborate religious cere- 
monies to aid the birth of the new moon, and 
performed sacrifices to save the sun during 
the crisis of an eclipse. There is no reason to 
believe that they were more stupid than we 
are. But at that epoch there had not been 
opportunity for the slow accumulation of 
clear and relevant ideas. 

The sort of way in which physical sciences 
grow into a form capable of treatment by 
mathematical methods is illustrated by the 
history of the gradual growth of the science 
of electromagnetism. Thunderstorms are 
events on a grand scale, arousing terror in 
men and even animals. From the earliest 
times they must have been objects of wild 



32 INTRODUCTION TO MATHEMATICS 

and fantastic hypotheses, though it may be 
doubted whether our modern scientific dis- 
coveries in connection with electricity are not 
more astonishing than any of the magical 
explanations of savages. The Greeks knew 
that amber (Greek, electron) when rubbed 
would attract light and dry bodies. In 
1600 A.D., Dr. Gilbert, of Colchester, published 
the first work on the subject in which any 
scientific method is followed. He made a 
list of substances possessing properties similar 
to those of amber ; he must also have the 
credit of connecting, however vaguely, electric 
and magnetic phenomena. At the end of the 
seventeenth and throughout the eighteenth 
century knowledge advanced. Electrical 
machines were made, sparks were obtained 
from them ; and the Leyden Jar was in- 
vented, by which these effects could be in- 
tensified. Some organised knowledge was 
being obtained ; but still no relevant mathe- 
matical ideas had been found out. Franklin, 
in the year 1752, sent a kite into the clouds 
and proved that thunderstorms were elec- 
trical. 

Meanwhile from the earliest epoch (2634 B.C.) 
the Chinese had utilized the characteristic 
property of the compass needle, but do not 
seem to have connected it with any theoretical 
ideas. The really profound changes in human 
life all have their ultimate origin in knowledge 



METHODS OF APPLICATION 33 

pursued for its own sake. The use of the com- 
pass was not introduced into Europe till the end 
of the twelfth century A.D., more than 3000 
years after its first use in China. The import- 
ance which the science of electromagnetism 
has since assumed in every department of 
human life is not due to the superior practical 
bias of Europeans, but to the fact that in the 
West electrical and magnetic phenomena 
were studied by men who were dominated by 
abstract theoretic interests. 

The discovery of the electric current is due 
to two Italians, Galvani in 1780, and Volta 
in 1792. This great invention opened a new 
series of phenomena for investigation. The 
scientific world had now three separate, 
though allied, groups of occurrences on hand 
the effects of " statical " electricity arising 
from frictional electrical machines, the mag- 
netic phenomena, and the effects due to 
electric currents. From the end of the 
eighteenth century onwards, these three lines 
of investigation were quickly inter-connected 
and the modern science of electromagnetism 
was constructed, which now threatens to 
transform human life. 

Mathematical ideas now appear. During 
the decade 1780 to 1789, Coulomb, a French- 
man, proved that magnetic poles attract or 
repel each other, in proportion to the inverse 
square of their distances, and also that the 



34 INTRODUCTION TO MATHEMATICS 

same law holds for electric charges laws 
curiously analogous to that of gravitation. 
In 1820, Oersted, a Dane, discovered that 
electric currents exert a force on magnets, 
and almost immediately afterwards the 
mathematical law of the force was correctly 
formulated by Ampere, a Frenchman, who 
also proved that two electric currents exerted 
forces on each other. " The experimental in- 
vestigation by which Ampere established the 
law of the mechanical action between electric 
currents is one of the most brilliant achieve- 
ments in science. The whole, theory and 
experiment, seems as if it had leaped, full- 
grown and full armed, from the brain of 
the ' Newton of Electricity.' It is perfect 
in form, and unassailable in accuracy, and it 
is summed up in a formula from which all 
the phenomena may be deduced, and which 
must always remain the cardinal formula of 
electro-dynamics." * 

The momentous laws of induction between 
currents and between currents and magnets 
were discovered by Michael Faraday in 1831- 
82. Faraday was asked: "What is the use 
of this discovery ? " He answered : " What is 
the use of a child it grows to be a man." 
Faraday's child has grown to be a man and 
is now the basis of all the modern applications 

* Electricity and Magnetism, Clerk Maxwell, VoL II., 
eh. iii. 



METHODS OF APPLICATION 35 

of electricity. Faraday also reorganized the 
whole theoretical conception of the science. 
His ideas, which had not been fully under- 
stood by the scientific world, were extended 
and put into a directly mathematical form by 
Clerk Maxwell in 1873. As a result of his 
mathematical investigations, Maxwell recog- 
nized that, under certain conditions, electrical 
vibrations ought to be propagated. He at 
once suggested that the vibrations which 
form light are electrical. This suggestion has 
since been verified, so that now the whole 
theory of light is nothing but a branch of the 
great science of electricity. Also Herz, a 
German, in 1888, following on Maxwell's 
ideas, succeeded in producing electric vibra- 
tions by direct electrical methods His 
experiments are the basis of our wireless 
telegraphy. 

In more recent years even more funda- 
mental discoveries have been made, and the 
science continues to grow in theoretic import- 
ance and in practical interest. This rapid 
sketch of its progress illustrates how, by the 
gradual introduction of the relevant theoretic 
ideas, suggested by experiment and them- 
selves suggesting fresh experiments, a whole 
mass of isolated and even trivial phenomena 
are welded together into one coherent science, 
in which the results of abstract mathematical 
deductions, starting from a few simple as- 



36 INTRODUCTION TO MATHEMATICS 

sumed laws, supply the explanation to the 
complex tangle of the course of events. 

Finally, passing beyond the particular 
sciences of electromagnetism and light, we 
can generalize our point of view still further, 
and direct our attention to the growth of 
mathematical physics considered as one great 
chapter of scientific thought. In the first 
place, what in the barest outlines is the story 
of its growth ? 

It did not begin as one science, or as the 
product of one band of men. The Chaldean 
shepherds watched the skies, the agents of 
Government in Mesopotamia and Egypt 
measured the land, priests and philosophers 
brooded on the general nature of all things. 
The vast mass of the operations of nature 
appeared due to mysterious unfathomable 
forces. " The wind bloweth where it listeth ? ' 
expresses accurately the blank ignorance then 
existing of any stable rules followed in detail 
by the succession of phenomena. In broad out- 
line, then as now, a regularity of events was 
patent. But no minute tracing of their inter- 
connection was possible, and there was no 
knowledge how even to set about to construct 
such a science. 

Detached speculations, a few happy or un- 
happy shots at the nature of things, formed 
the utmost which could be produced. 

Meanwhile land-surveys had produced geo- 



METHODS OF APPLICATION 87 

metry, and the observations of the heavens 
disclosed the exact regularity of the solar 
system. Some of the later Greeks, such as 
Archimedes, had just views on the elementary 
phenomena of hydrostatics and optics. In- 
deed, Archimedes, who combined a genius for 
mathematics with a physical insight, must 
rank with Newton, who lived nearly two 
thousand years later, as one of the founders 
of mathematical physics. He lived at Syra- 
cuse, the great Greek city of Sicily. When 
the Romans besieged the town (in 212 to 
210 B.C.), he is said to have burned their ships 
by concentrating on them, by means of 
mirrors, the sun's rays. The story is highly 
improbable, but is good evidence of the repu- 
tation which he had gained among his con- 
temporaries for his knowledge of optics. At 
the end of this siege he was killed. According 
to one account given by Plutarch, in his life of 
Marcellus, he was found by a Roman soldier 
absorbed in the study of a geometrical diagram 
which he had traced on the sandy floor of his 
room. He did not immediately obey the orders 
of his captor, and so was killed. For the credit 
of the Roman generals it must be said that 
the soldiers had orders to spare him. The 
internal evidence for the other famous story 
of him is very strong ; for the discovery 
attributed to him is one eminently worthy of 
his genius for mathematical and physical re- 



38 INTRODUCTION TO MATHEMATICS 

search. Luckily, it is simple enough to be 
explained here in detail. It is one of the best 
easy examples of the method of application 
of mathematical ideas to physics. 

Hiero, King of Syracuse, had sent a quan- 
tity of gold to some goldsmith to form the 
material of a crown. He suspected that the 
craftsmen had abstracted some of the gold 
and had supplied its place by alloying the 
remainder with some baser metal. Hiero 
sent the crown to Archimedes and asked him 
to test it. In these days an indefinite num- 
ber of chemical tests would be available. 
But then Archimedes had to think out the 
matter afresh. The solution flashed upon 
him as he lay in his bath. He jumped 
up and ran through the streets to the 
palace, shouting Eureka! Eureka! (I have 
found it, I have found it). This day, if we 
knew which it was, ought to be celebrated as 
the birthday of mathematical physics ; the 
science came of age when Newton sat in his 
orchard. Archimedes had in truth made a 
great discovery. He saw that a body when 
immersed in water is pressed upwards by the 
surrounding water with a resultant force 
equal to the weight of the water it displaces. 
This law can be proved theoretically from the 
mathematical principles of hydrostatics and 
can also be verified experimentally. Hence, 
if W Ib. be the weight of the crown, as weighed 



METHODS OF APPLICATION 39 

in air, and w Ib. be the weight of the water 
which it displaces when completely immersed, 
W w would be the extra upward force 
necessary to sustain the crown as it hung in 
water. 

Now, this upward force can easily be ascer- 
tained by weighing the body as it hangs in 
water, as shown in the annexed figure. If 



The 

crown 



Weights 



Fig. 3. 

the weights in the right-hand scale come to 
F Ib., then the apparent weight of the crown 
in water is F Ib. ; and we thus have 

F=W-w 



and thus 
and 



W 

w 



W 



W-F 

where W and F are determined by the easy, 
and fairly precise, operation of weighing. 



40 INTRODUCTION TO MATHEMATICS 

W 

Hence, by equation (A), is known. But 

W 

is the ratio of the weight of the crown to 

to 

the weight of an equal volume of water. 
This ratio is the same for any lump of metal of 
the same material : it is now called the specific 
gravity of the material, and depends only on 
the intrinsic nature of the substance and not 
on its shape or quantity. Thus to test if the 
crown were of gold, Archimedes had only to 
take a lump of indisputably pure gold and 
find its specific gravity by the same process. 
If the two specific gravities agreed, the crown 
was pure ; if they disagreed, it was debased. 

This argument has been given at length, 
because not only is it the first precise example 
of the application of mathematical ideas to 
physics, but also because it is a perfect and 
simple example of what must be the method 
and spirit of the science for all time. 

The death of Archimedes by the hands of a 
Roman soldier is symbolical of a world-change 
of the first magnitude : the theoretical Greeks, 
with their love of abstract science, were super- 
seded in the leadership of the European world 
by the practical Romans. Lord Beacons- 
field, in one of his novels, has defined a practi- 
cal man as a man who practises the errors of 
his forefathers. The Romans were a great 
race, but they were cursed with the sterility 



METHODS OF APPLICATION 41 

which waits upon practicality. They did not 
improve upon the knowledge of their fore- 
fathers, and all their advances were confined 
to the minor technical details of engineering. 
They were not dreamers enough to arrive at 
new points of view, which could give a more 
fundamental control over the forces of nature. 
No Roman lost his life because he was ab- 
sorbed in the contemplation of a mathe- 
matical diagram. 



CHAPTER IV 

DYNAMICS 

THE world had to wait for eighteen hundred 
years till the Greek mathematical physicists 
found successors. In the sixteenth and seven- 
teenth centuries of our era great Italians, in 
particular Leonardo da Vinci, the artist 
(born 1452, died 1519), and Galileo (born 1564, 
died 1642), rediscovered the secret, known to 
Archimedes, of relating abstract mathematical 
ideas with the experimental investigation of 
natural phenomena. Meanwhile the slow 
advance of mathematics and the accumula- 
tion of accurate astronomical knowledge had 
placed natural philosophers in a much more 
advantageous position for research. Also the 
very egoistic self-assertion of that age, its 
greediness for personal experience, led its 
thinkers to want to see for themselves what 
happened ; and the secret of the relation of 
mathematical theory and experiment in in- 
ductive reasoning was practically discovered. 
It was an act eminently characteristic of the 
age that Galileo, a philosopher, should have 

42 



DYNAMICS 43 

dropped the weights from the leaning tower 
of Pisa. There are always men of thought 
and men of action ; mathematical physics is 
the product of an age which combined in the 
same men impulses to thought with impulses 
to action. 

This matter of the dropping of weights from 
the tower marks picturesquely an essential 
step in knowledge, no less a step than the 
first attainment of correct ideas on the science 
of dynamics, the basal science of the whole 
subject. The particular point in dispute was 
as to whether bodies of different weights 
would fall from the same height in the same 
time. According to a dictum of Aristotle, 
universally followed up to that epoch, the 
heavier weight would fall the quicker. Gali- 
leo affirmed that they would fall in the same 
time, and proved his point by dropping 
weights from the top of the leaning tower. 
The apparent exceptions to the rule all arise 
when, for some reason, such as extreme light- 
ness or great speed, the air resistance is im- 
portant. But neglecting the air the law is 
exact. 

Galileo's successful experiment was not the 
result of a mere lu'cky guess. It arose from 
his correct ideas in connection with inertia 
and mass. The first law of motion, as follow- 
ing Newton we now enunciate it, is Every 
body continues in its state of rest or of uni- 



44 INTRODUCTION TO MATHEMATICS 

form motion in a straight line, except so far 
as it is compelled by impressed force to 
change that state. This law is more than a 
dry formula : it is also a paean of triumph 
over defeated heretics. The point at issue 
can be understood by deleting from the law 
the phrase " or of uniform motion in a straight 
line." We there obtain what might be taken 
as the Aristotelian opposition formula: 
** Every body continues in its state of rest 
except so far as it is compelled by impressed 
force to change that state." 

In this last false formula it is asserted that, 
apart from force, a body continues in a state 
of rest ; and accordingly that, if a body is 
moving, a force is required to sustain the 
motion ; so that when the force ceases, the 
motion ceases. The true Newtonian law 
takes diametrically the opposite point of view. 
The state of a body unacted on by force is 
that of uniform motion in & straight line, and 
no external force or influence is to be looked 
for as the cause, or, if you like to put it so, as 
the invariable accompaniment of this uniform 
rectilinear motion. Rest is merely a par- 
ticular case of such motion, merely when the 
velocity is and remains zero. Thus, when a 
body is moving, we do not seek for any ex- 
ternal influence except to explain changes in 
the rate of the velocity or changes in its direc- 
tion. So long as the body is moving at the 



DYNAMICS 45 

same rate and in the same direction there is 
no need to invoke the aid of any forces. 

The difference between the two points of 
view is well seen by reference to the theory of 
the motion of the planets. Copernicus, a 
Pole, born at Thorn in West Prussia (born 
1473, died 1543), showed how much simpler 
it was to conceive the planets, including the 

Force (on False hypothesis) 




Fig. 4. 

earth as revolving round the sun in orbits 
which are nearly circular ; and later, Kepler, 
a German mathematician, in the year 1609 
proved that, in fact, the orbits are practically 
ellipses, that is, a special sort of oval curves 
which we will consider later in more detail. 
Immediately the question arose as to what 
are the forces which preserve the planets in 
this motion. According to the old false view, 



46 INTRODUCTION TO MATHEMATICS 

held by Kepler, the actual velocity itself re- 
quired preservation by force. Thus he looked 
for tangential forces as in the accompanying 
figure (4). But according to the Newtonian 
law, apart from some force the planet would 
move for ever with its existing velocity in a 
straight line, and thus depart entirely from 
the sun. Newton, therefore, had to search 
for a force which would bend the motion 

Planer 




Fig. 5. 

round into its elliptical orbit. This he showed 
must be a force directed towards the sun as in 
the next figure (5). In fact, the force is the 
gravitational attraction of the sun acting 
according to the law of the inverse square of 
the distance, which has been stated above. 

The science of mechanics rose among the 
Greeks from a consideration of the theory of 
the mechanical advantage obtained by the use 



DYNAMICS 47 

of a lever, and also from a consideration of 
various problems connected with the weights 
of bodies. It was finally put on its true basis 
at the end of the sixteenth and during the 
seventeenth centuries, as the preceding ac- 
count shows, partly with the view of explain- 
ing the theory of falling bodies, but chiefly 
in order to give a scientific theory of planetary 
motions. But since those days dynamics has 
taken upon itself a more ambitious task, and 
now claims to be the ultimate science of which 
the others are but branches. The claim 
amounts to this : namely, that the various 
qualities of things perceptible to the senses 
are merely our peculiar mode of appreciating 
changes in position on the part of things 
existing in space. For example, suppose we 
look at Westminster Abbey. It has been 
standing there, grey and immovable, for cen- 
turies past. But, according to modern scien- 
tific theory, that greyness, which so heightens 
our sense of the immobility of the building, is 
itself nothing but our way of appreciating the 
rapid motions of the ultimate molecules, which 
form the outer surface of the building and 
communicate vibrations to a substance called 
the ether. Again we lay our hands on its 
stones and note their cool, even temperature, 
so symbolic of the quiet repose of the building. 
But this feeling of temperature simply marks 
our sense of the transfer of heat from the 



48 INTRODUCTION TO MATHEMATICS 

hand to the stone, or from the stone to the 
hand ; and, according to modern science, 
heat is nothing but the agitation of the mole- 
cules of a body. Finally, the organ begins 
playing, and again sound is nothing but the 
result of motions of the air striking on the 
drum of the ear. 

Thus the endeavour to give a dynamical 
explanation of phenomena is the attempt to 
explain them by statements of the general 
form, that such and such a substance or body 
was in this place and is now in that place. 
Thus we arrive at the great basal idea of 
modern science, that all our sensations are 
the result of comparisons of the changed 
configurations of things in space at various 
times. It follows therefore, that the laws 
of motion, that is, the laws of the changes 
of configurations of things, are the ultimate 
laws of physical science. 

In the application of mathematics to the 
investigation of natural philosophy, science 
does systematically what ordinary thought 
does casually. When we talk of a chair, we 
usually mean something which we have been 
seeing or feeling in some way ; though most 
of our language will presuppose that there 
is something which exists independently of 
our sight or feeling. Now in mathematical 
physics the opposite course is taken. The 
chair is conceived without any reference to 



DYNAMICS 49 

anyone in particular, or to any special modes 
of perception. The result is that the chair 
becomes in thought a set of molecules in space, 
or a group of electrons, a portion of the ether 
in motion, or however the current scientific 
ideas describe it. But the point is that 
science reduces the chair to things moving in 
space and influencing each other's motions. 
Then the various elements or factors which 
enter into a set of circumstances, as thus 
conceived, are merely the things, like lengths 
of lines, sizes of angles, areas, and volumes, by 
which the positions of bodies in space can be 
settled. Of course, in addition to these geo- 
metrical elements the fact of motion and 
change necessitates the introduction of the 
rates of changes of such elements, that is to 
say, velocities, angular velocities, accelera- 
tions, and suchlike things. Accordingly, mathe- 
matical physics deals with correlations be- 
tween variable numbers which are supposed 
to represent the correlations which exist in 
nature between the measures of these geo- 
metrical elements and of their rates of change. 
But always the mathematical laws deal with 
variables, and it is only in the occasional 
testing of the laws by reference to experi- 
ments, or in the use of the laws for special 
predictions that definite numbers are substi- 
tuted. 
The interesting point about the world as 



50 INTRODUCTION TO MATHEMATICS 

thus conceived in this abstract way through- 
out the study of mathematical physics, where 
only the positions and shapes of things are 
considered together with their changes, is that 
the events of such an abstract world are suffi- 
cient to "explain" our sensations. When we 
hear a sound, the molecules of the air have 
been agitated in a certain way : given the 
agitation, or air- waves as they are called, all 
normal people hear sound ; and if there are 
no air-waves, there is no sound. And, simi- 
larly, a physical cause or origin, or parallel 
event (according as different people might like 
to phrase it) underlies our other sensations. 
Our very thoughts appear to cprrespond to 
conformations and motions of the brain ; in- 
jure the brain and you injure the thoughts. 
Meanwhile the events of this physical universe 
succeed each other according to the mathe- 
matical laws which ignore all special sensa- 
tions and thoughts and emotions. 

Now, undoubtedly, this is the general aspect 
of the relation of the world of mathematical 
physics to our emotions, sensations, and 
thoughts ; and a great deal of controversy 
has been occasioned by it and much ink 
spilled. We need only make one remark. The 
whole situation has arisen, as we have seen, 
from the endeavour to describe an external 
world " explanatory " of our various in- 
dividual sensations and emotions, but a world 



DYNAMICS 51 

also, not essentially dependent upon any 
particular sensations or upon any particular 
individual. Is such a world merely but 
one huge fairy tale ? But fairy tales are 
fantastic and arbitrary : if in truth there 
be such a world, it ought to submit itself 
to an exact description, which determines 
accurately its various parts and their mutual 
relations. Now, to a large degree, this 
scientific world does submit itself to this 
test and allow its events to be explored 
and predicted by the apparatus of abstract 
mathematical ideas. It certainly seems that 
here we have an inductive verification of 
our initial assumption. It must be admitted 
that no inductive proof is conclusive ; but 
if the whole idea of a world which has 
existence independently of our particular per- 
ceptions of it be erroneous, it requires careful 
explanation why the attempt to characterise 
it, in terms of that mathematical remnant 
of our ideas which would apply to it, should 
issue in such a remarkable success. 

It would take us too far afield to enter into 
a detailed explanation of the other laws of 
motion. The remainder of this chapter must 
be devoted to the explanation of remarkable 
ideas which are fundamental, both to mathe- 
matical physics and to pure mathematics : 
these are the ideas of vector quantities and 
the parallelogram law for vector addition. We 



52 INTRODUCTION TO MATHEMATICS 

have seen that the essence of motion is that 
a body was at A and is now at C. This trans- 
ference from A to C requires two distinct 
elements to be settled before it is completely 
determined, namely its magnitude (i.e. the 
length AC) and its direction. Now any- 
thing, like this transference, which is com- 
pletely given by the determination of a magni- 




tude and a direction is called a vector. For 
example, a velocity requires for its definition 
the assignment of a magnitude and of a 
direction. It must be of so many miles per 
hour in such and such a direction. The ex- 
istence and the independence of these two 
elements in the determination of a velocity 
are well illustrated by the action of the captain 
of a ship, who communicates with different sub- 
ordinates respecting them : he tells the chief 
engineer the number of knots at which he is 
to steam, and the helmsman the compass 



DYNAMICS 58 

bearing of the course which he is to keep. 
Again the rate of change of velocity, that is 
velocity added per unit time, is also a vector 
quantity : it is called the acceleration. Simi- 
larly a force in the dynamical sense is another 
vector quantity. Indeed, the vector nature 
of forces follows at once according to dynami- 
cal principles from that of velocities and 
accelerations ; but this is a point which we 
need not go into. It is sufficient here to say 
that a force acts on a body with a certain 
magnitude in a certain direction. 

Now all vectors can be graphically repre- 
sented by straight lines. All that has to be 
done is to arrange : (i) a scale according to 
which units of length correspond to units of 
magnitude of the vector for example, one 
inch to a velocity of 10 miles per hour in the 
case of velocities, and one inch to a force of 
10 tons weight in the case of forces and (ii) 
a direction of the line on the diagram corre- 
sponding to the direction of the vector. Then 
a line drawn with the proper number of inches 
of length in the proper direction represents the 
required vector on the arbitrarily assigned scale 
of magnitude. This diagrammatic representa- 
tion of vectors is of the first importance. By 
its aid we can enunciate the famous " parallelo- 
gram law " for the addition of vectors of the 
same kind but in different directions. 

Consider the vector AC in figure 6 as repre- 



54 INTRODUCTION TO MATHEMATICS 

sentative of the changed position of a body 
from A to C : we will call this the vector of 
transportation. It will be noted that, if the 
reduction of physical phenomena to mere 
changes in positions, as explained above, is 
correct, all other types of physical vectors are 
really reducible in some way or other to this 
single type. Now the final transportation 
from A to C is equally well effected by a 
transportation from A to B and a transporta- 
tion from B to C, or, completing the parallelo- 
gram ABCD, by a transportation from A to 
D and a transportation from D to C. These 
transportations as thus successively applied 
are said to be ad<ded together. This is simply 
a definition of what we mean by the addition 
of transportations. Note further that, con- 
sidering parallel lines as being lines drawn in 
the same direction, the transportations B to 
C and A to D may be conceived as the same 
transportation applied to bodies in the two 
initial positions B and A. With this con- 
ception we may talk of the transportation 
A to D as applied to a body in any position, 
for example at B. Thus we may say that 
the transportation A to C can be conceived 
as the sum of the two transportations A to 
B and A to D applied in any order. Here 
we have the parallelogram law for the ad- 
dition of transportations : namely, if the 
transportations are A to B and A to D, 



DYNAMICS 55 

complete the parallelogram ABCD, and then 
the sum of the two is the diagonal AC. 

All this at first sight may seem to be 
very artificial. But it must be observed 
that nature itself presents us with the idea. 
For example, a steamer is moving in the 
direction AD (cf. fig. 6) and a man walks 
across its deck. If the steamer were still, 
in one minute he would arrive at B ; but 
during that minute his starting point A on 
the deck has moved to D, and his path on 
the deck has moved from AB to DC. So 
that, in fact, his transportation has been from 
A to C over the surface of the sea. It is, 
however, presented to us analysed into the 
sum of two transportations, namely, one from 
A to B relatively to the steamer, and one 
from A to D which is the transportation of 
the steamer. 

By taking into account the element of time, 
namely one minute, this diagram of the man's 
transportation AC represents his velocity. 
For if AC represented so many feet of trans- 
portation, it now represents a transportation 
of so many feet per minute, that is to say, it 
represents the velocity of the man. Then 
AB and AD represent two velocities, namely, 
his velocity relatively to the steamer, and the 
velocity of the steamer, whose "sum" makes 
up his complete velocity. It is evident that 
diagrams and definitions concerning trans- 



56 INTRODUCTION TO MATHEMATICS 

portations are turned into diagrams and de- 
finitions concerning velocities by conceiving 
the diagrams as representing transportations 
per unit time. Again, diagrams and defini- 
tions concerning velocities are turned into 
diagrams and definitions concerning accelera- 




Fig. 7. 

tions by conceiving the diagrams as repre- 
senting velocities added per unit time. 

Thus by the addition of vector velocities 
and of vector accelerations, we mean the 
addition according to the parallelogram law. 

Also, according to the laws of motion a 
force is fully represented by the vector 
acceleration it produces in a body of given 
mass. Accordingly, forces will be said to be 
added when their joint effect is to be reckoned 
according to the parallelogram law. 

Hence for the fundamental vectors of 



DYNAMICS 57 

science, namely transportations, velocities, 
and forces, the addition of any two of the same 
kind is the production of a " resultant " 
vector according to the rule of the parallelo- 
gram law. 

By far the simplest type of parallelogram 
is a rectangle, and in pure mathematics it is 
the relation of the single vector AC to the 
two component vectors, AB and AD, at right 
angles (cf. fig. 7), which is continually re- 
curring. Let x, y, and r units represent the 
lengths of AB, AD, and AC, and let m units 
of angle represent the magnitude of the angle 
BAG. Then the relations between #, y, r, 
and m, in all their many aspects are the con- 
tinually recurring topic of pure mathematics ; 
and the results are of the type required for 
application to the fundamental vectors of 
mathematical physics. This diagram is the 
chief bridge over which the results of pure 
mathematics pass in order to obtain applica- 
tion to the facts of nature. 



CHAPTER V 

THE SYMBOLISM OF MATHEMATICS 

WE now return to pure mathematics, and 
consider more closely the apparatus of ideas 
out of which the science is built. Our first 
concern is with the symbolism of the science, 
and we start with the simplest and universally 
known symbols, namely those of arithmetic. 

Let us assume for the present that we have 
sufficiently clear ideas about the integral 
numbers, represented in the Arabic notation 
by 0,1,2, . . ., 9, 10, 11, ... 100, 101, . . . and 
so on. This notation was introduced into 
Europe through the Arabs, but they appar- 
ently obtained it from Hindoo sources. The 
first known work * in which it is systematic- 
ally explained is a work by an Indian mathe- 
matician, Bhaskara (born 1114 A.D.). But 
the actual numerals can be traced back to the 
seventh century of our era, and perhaps were 
originally invented in Tibet. For our present 

* For the detailed historical facts relating to pure 
mathematics, I am chiefly indebted to A Short History 
of Mathematics, by W. W. B. Ball. 

58 



SYMBOLISM OF MATHEMATICS 59 

purposes, however, the history of the notation 
is a detail. The interesting point to notice 
is the admirable illustration which this 
numeral system affords of the enormous im- 
portance of a good notation. By relieving 
the brain of all unnecessary work, a good 
notation sets it free to concentrate on more 
advanced problems, and in effect increases 
the mental power of the race. Before the 
introduction of the Arabic notation, multipli- 
cation was difficult, and the division even of 
integers called into play the highest mathe- 
matical faculties. Probably nothing in the 
modern world would have more astonished a 
Greek mathematician than to learn that, under 
the influence of compulsory education, a 
large proportion of the population of Western 
Europe could perform the operation of 
division for the largest numbers. This fact 
would have seemed to him a sheer impos- 
sibility. The consequential extension of 
the notation to decimal fractions was not 
accomplished till the seventeenth century. 
Our modern power of easy reckoning with 
decimal fractions is the almost miraculous 
result of the gradual discovery of a perfect 
notation. 

Mathematics is often considered a diffi- 
cult and mysterious science, because of the 
numerous symbols which it employs. Of 
course, nothing is more incomprehensible than 



60 INTRODUCTION TO MATHEMATICS 

a symbolism which we do not understand. 
Also a symbolism, which we only partially 
understand and are unaccustomed to use, is 
difficult to follow. In exactly the same way 
the technical terms of any profession or trade 
are incomprehensible to those who have never 
been trained to use them. But this is not 
because they are difficult in themselves. On 
the contrary they have invariably been intro- 
duced to make things easy. So in mathe- 
matics, granted that we are giving any serious 
attention to mathematical ideas, the sym- 
bolism is invariably an immense simplifica- 
tion. It is not only of practical use, but is 
of great interest. For it represents an analy- 
sis of the ideas of the subject and an almost 
pictorial representation of their relations to 
each other. If anyone doubts the utility of 
symbols, let him write out in full, without any 
symbol whatever, the whole meaning of the 
following equations which represent some of 
the fundamental laws of algebra * : 

z+y=y+x (l) 

(x+y}+z=x+(y+z) .. . . (2) 

as x y=y xx (3) 

(x x y) x z=x x (y x z) . . (4) 
x x (y+z)=(x x y)+(x x z) . . (5) 
Here (1) and (2) are called the commutative 
and associative laws for addition, (3) and (4) 

* Cf. Note A, p. 250. 



SYMBOLISM OF MATHEMATICS 61 

are the commutative and associative laws for 
multiplication, and (5) is the distributive law 
relating addition and multiplication. For ex- 
ample, without symbols, (1) becomes: If a 
second number be added to any given number 
the result is the same as if the first given 
number had been added to the second number. 

This example shows that, by the aid of sym- 
bolism, we can make transitions in reasoning 
almost mechanically by the eye, which other- 
wise would call into play the higher faculties 
of the brain. 

It is a profoundly erroneous truism, repeated 
by all copy-books and by eminent people when 
they are making speeches, that we should 
cultivate the habit of thinking of what we are 
doing. The precise opposite is the case. 
Civilization advances by extending the num- 
ber of important operations which we can 
perform without thinking about them. Opera- 
tions of thought are like cavalry charges in 
a battle they are strictly limited in num- 
ber, they require fresh horses, and must only 
be made at decisive moments. 

One very important property for symbolism 
to possess is that it should be concise, so as to 
be visible at one glance of the eye and to be 
rapidly written. Now we cannot place sym- 
bols more concisely together than by placing 
them in immediate juxtaposition. In a good 
symbolism therefore, the juxtaposition of im- 



02 INTRODUCTION TO MATHEMATICS 

portant symbols should have an important 
meaning. This is one of the merits of the 
Arabic notation for numbers ; by means of 
ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and by 
simple juxtaposition it symbolizes any number 
whatever. Again in algebra, when we have 
two variable numbers x and y, we have to 
make a choice as to what shall be denoted by 
their juxtaposition xy. Now the two most 
important ideas on hand are those of addition 
and multiplication. Mathematicians have 
chosen to make their symbolism more concise 
by denning xy to stand for x x y. Thus the 
laws (3), (4), and (5) above are in general 
written, 

xy=yx, (xy}z=x(yz) y x(y+z)=xy+xz, 
thus securing a great gain in conciseness. 
The same rule of symbolism is applied to the 
juxtaposition of a definite number and a vari- 
able : we write 3x for 3 x x, and 30x for 30 x x. 

It is evident that in substituting definite 
numbers for the variables some care must be 
taken to restore the x, so as not to conflict 
with the Arabic notation. Thus when we 
substitute 2 for x and 3 for y in xy, we must 
write 2x3 for xy, and not 23 which means 
20+3. 

It is interesting to note how important for 
the development of science a modest-looking 
symbol may be. It may stand for the em- 
phatic presentation of an idea, often a very 



SYMBOLISM OF MATHEMATICS 63 

subtle idea, and by its existence make it easy 
to exhibit the relation of this idea to all the 
complex trains of ideas in which it occurs. 
For example, take the most modest of all 
symbols, namely, 0, which stands for the num- 
ber zero. The Roman notation for numbers 
had no symbol for zero, and probably most 
mathematicians of the ancient world would 
have been horribly puzzled by the idea of the 
number zero. For, after all, it is a very 
subtle idea, not at all obvious. A great deal 
of discussion on the meaning of the zero of 
quantity will be found in philosophic works. 
Zero is not, in real truth, more difficult or 
subtle in idea than the other cardinal numbers. 
What do we mean by 1 or by 2, or by 3 ? 
But we are familiar with the use of these ideas, 
though we should most of us be puzzled to 
give a clear analysis of the simpler ideas 
which go to form them. The point about zero 
is that we do not need to use it in the opera- 
tions of daily life. No one goes out to buy 
zero fish. It is in a way the most civilized 
of all the cardinals, and its use is only forced 
on us by the needs of cultivated modes of 
thought. Many important services are ren- 
dered by the symbol 0, which stands for the 
number zero. 

The symbol developed in connection with 
the Arabic notation for numbers of which it 
is an essential part. For in that notation the 



64 INTRODUCTION TO MATHEMATICS 

value of a digit depends on the position in 
which it occurs. Consider, for example, the 
digit 5, as occurring in the numbers 25, 51, 
3512, 5213. In the first number 5 stands for 
five, in the second number 5 stands for fifty, 
in the third number for five hundred, and in 
the fourth number for five thousand. Now, 
when we write the number fifty-one in the 
symbolic form 51, the digit 1 pushes the digit 
5 along to the second place (reckoning from 
right to left) and thus gives it the value fifty. 
But when we want to symbolize fifty by itself, 
we can have no digit 1 to perform this service ; 
we want a digit in the units place to add 
nothing to the total and yet to push the 5 
along to the second place. This service is 
performed by 0, the symbol for zero. It is 
extremely probable that the men who intro- 
duced for this purpose had no definite con- 
ception in their minds of the number zero. 
They simply wanted a mark to symbolize the 
fact that nothing was contributed by the 
digit's place in which it occurs. The idea of 
zero probably took shape gradually from a 
desire to assimilate the meaning of this mark 
to that of the marks, 1, 2, ... 9, which do re- 
present cardinal numbers. This would not 
represent the only case in which a subtle idea 
has been introduced into mathematics by a 
symbolism which in its origin was dictated by 
practical convenience. 



SYMBOLISM OF MATHEMATICS 65 

Thus the first use of was to make the 
arable notation possible no slight service. 
We can imagine that when it had been intro- 
duced for this purpose, practical men, of the 
sort who dislike fanciful ideas, deprecated the 
silly habit of identifying it with a number 
zero. But they were wrong, as such men 
always are when they desert their proper 
function of masticating food which others have 
prepared. For the next service performed by 
the symbol essentially depends upon assign- 
ing to it the function of representing the 
number zero. 

This second symbolic use is at first sight 
so absurdly simple that it is difficult to make 
a beginner realize its importance. Let us 
start with a simple example. In Chapter II. 
we mentioned the correlation between two 
variable numbers x and y represented by the 
equation x -\-y 1. This can be represented 
in an indefinite number of ways ; for example, 
x = 1 y, y= Ix, 2x+3y 1 = x-\-2y, and so 
on. But the important way of stating it is 

x+y-I = 0. 

Similarly the important way of writing the 
equation x=I is a? 1=0, and of representing 
the equation 3x 2=2x* is 2x* 30+2=0. 
The point is that all the symbols which repre- 
sent variables, e.g. x and y, and the symbols 



66 INTRODUCTION TO MATHEMATICS 

representing some definite number other than 
zero, such as 1 or 2 in the examples above, 
are written on the left-hand side, so that the 
whole left-hand side is equated to the number 
zero. The first man to do this is said to 
have been Thomas Harriot, born at Oxford 
in 1560 and died in 1621. But what is the 
importance of this simple symbolic pro- 
cedure ? It made possible the growth of the 
modern conception of algebraic form. 

This is an idea to which we shall have con- 
tinually to recur ; it is not going too far to 
say that no part of modern mathematics can 
be properly understood without constant re- 
currence to it. The conception of form is 
so general that it is difficult to characterize 
it in abstract terms. At this stage we shall 
do better merely to consider examples. Thus 
the equations 2x 3=0, #1=0, 5x 6=0, 
are all equations of the same form, namely, 
equations involving one unknown x, which is 
not multiplied by itself, so that a? 2 , a? 3 , etc., do 
not appear. Again 3a? 2 2x + 1 = 0, x 2 3x + 2 
=0, o? 2 4=0, are all equations of the same 
form, namely, equations involving oneunknown 
x in which a?xa;, that is a? 2 , appears. These 
equations are called quadratic equations. 
Similarly cubic equations, in which a? 3 appears, 
yield another form, and so on. Among the 
three quadratic equations given above there 
is a minor difference between the last equa- 



SYMBOLISM OF MATHEMATICS 67 

tion, x 2 4=0, and the preceding two equa- 
tions, due to the fact that x (as distinct 
from x 2 ) does not appear in the last and 
does in the other two. This distinction is 
very unimportant in comparison with the 
great fact that they are all three quadratic 
equations. 

Then further there are the forms of equation 
stating correlations between two variables; 
for example, x+y 1=0, 2x -\-3y- 8=0, and 
so on. These are examples of what is called 
the linear form of equation. The reason for 
this name of " linear " is that the graphic 
method of representation, which is explained 
at the end of Chapter II, always represents 
such equations by a straight line. Then there 
are other forms for two variables for example, 
the quadratic form, the cubic form, and so on. 
But the point which we here insist upon is 
that this study of form is facilitated, and, 
indeed, made possible, by the standard method 
of writing equations with the symbol on 
the right-hand side. 

There is yet another function performed by 
in relation to the study of form. Whatever 
number x may be, x x=Q, and #+0=#. 
By means of these properties minor differ- 
ences of form can be assimilated. Thus the 
difference mentioned above between the quad- 
ratic equations x 2 3#-f-2=0, and x 2 4=0, 
can be obliterated by writing the latter 



68 INTRODUCTION TO MATHEMATICS 

equation in the form # 2 +(Oxo:) 4=0. For, 
by the laws stated above, # 2 +(Ox#) 4 = 
a^+O 4=# 2 4. Hence the equation # 2 4 
=0, is merely representative of a particular 
class of quadratic equations and belongs to 
the same general form as does x 2 3#-f2=0. 

For these three reasons the symbol 0, re- 
presenting the number zero, is essential to 
modern mathematics. It has rendered pos- 
sible types of investigation which would have 
been impossible without it. 

The symbolism of mathematics is in truth 
the outcome of the general ideas which 
dominate the science. We have now two 
such general ideas before us, that of the vari- 
able and that of algebraic form. The junction 
of these concepts has imposed on mathematics 
another type of symbolism almost quaint in 
its character, but none the less effective. We 
have seen that an equation involving two 
variables, x and y, represents a particular 
correlation between the pair of variables. 
Thus x -{-y 1 =0 represents one definite corre- 
lation, and 3x+2y 5=0 represents another 
definite correlation between the variables x 
and y ; and both correlations have the form 
of what we have called linear correlations. 
But now, how can we represent any linear 
correlation between the variable numbers x 
and y ? Here we want to symbolize any 
linear correlation ; just as x symbolizes any 



SYMBOLISM OF MATHEMATICS 69 

number. This is done by turning the numbers 
which occur in the definite correlation 3x+2y 
5 =o into letters. We obtain ax -\-by c =0. 
Here a, b, c, stand for variable numbers just 
as do x and y : but there is a difference in the 
use of the two sets of variables. We study 
the general properties of the relationship be- 
tween x and y while a, b, and c have un- 
changed values. We do not determine what 
the values of a, b, and c are ; but whatever 
they are, they remain fixed while we study 
the relation between the variables x and y 
for the whole group of possible values of x 
and y. But when we have obtained the pro- 
perties of this correlation, we note that, be- 
cause a, b, and c have not in fact been deter- 
mined, we have proved properties which must 
belong to any such relation. Thus, by now 
varying a, b, and c, we arrive at the idea that 
ax+by c=0 represents a variable linear 
correlation between x and y. In comparison 
with x and t/, the three variables a, b, and c 
are called constants. Variables used in this 
way are sometimes also called parameters. 

Now, mathematicians habitually save the 
trouble of explaining which of their variables 
are to be treated as " constants," and which 
as variables, considered as correlated in their 
equations, by using letters at the end of the 
alphabet for the " variable " variables, and 
letters at the beginning of the alphabet for 



70 INTRODUCTION TO MATHEMATICS 

the " constant " variables, or parameters. 
The two systems meet naturally about the 
middle of the alphabet. Sometimes a word 
or two of explanation is necessary ; but as a 
matter of fact custom and common sense are 
usually sufficient, and surprisingly little con- 
fusion is caused by a procedure which seems 
so lax. 

The result of this continual elimination of 
definite numbers by successive layers of para- 
meters is that the amount of arithmetic per- 
formed by mathematicians is extremely small. 
Many mathematicians dislike all numerical 
computation and are not particularly expert 
at it. The territory of arithmetic ends where 
the two ideas of "variables" and of "alge- 
braic form " commence their sway. 



CHAPTER VI 

GENERALIZATIONS OT NUMBER 

ONE great peculiarity of mathematics is the 
set of allied ideas which have been invented 
in connection with the integral numbers from 
which we started. These ideas may be called 
extensions or generalizations of number. In 
the first place there is the idea of fractions. 
The earliest treatise on arithmetic which we 
possess was written by an Egyptian priest, 
named Ahmes, between 1700 B.C. and 1100 
B.C., and it is probably a copy of a much older 
work. It deals largely with the properties of 
fractions. It appears, therefore, that this 
concept was developed very early in the his- 
tory of mathematics. Indeed the subject is 
a very obvious one. To divide a field into 
three equal parts, and to take two of the 
parts, must be a type of operation which had 
often occurred. Accordingly, we need not be 
surprised that the men of remote civilizations 
were familiar with the idea of two-thirds, and 

71 



72 INTRODUCTION TO MATHEMATICS 

with allied notions. Thus as the first genera- 
lization of number we place the concept of 
fractions. The Greeks thought of this sub- 
ject rather in the form of ratio, so that a 
Greek would naturally say that a line of 
two feet in length bears to a line of three 
feet in length the ratio of 2 to 8. Under 
the influence of our algebraic notation we 
would more often say that one line was 
two-thirds of the other in length, and would 
think of two-thirds as a numerical mul- 
tiplier. 

In connection with the theory of ratio, or 
fractions, the Greeks made a great discovery, 
which has been the occasion of a large amount 
of philosophical as well as mathematical 
thought. They found out the existence of 
" incommensurable " ratios. They proved, 
in fact, during the course of their geometrical 
investigations that, starting with a line of any 
length, other lines must exist whose lengths 
do not bear to the original length the ratio 
of any pair of integers or, in other words, 
that lengths exist which are not any exact 
fraction of the original length. 

For example, the diagonal of a square cannot 
be expressed as any fraction of the side of the 
same square ; in our modern notation the 
length of the diagonal is \/2 times the length 
of the side. But there is no fraction which 
exactly represents \/2. We can approximate 



GENERALIZATIONS OF NUMBERS 73 

to V2 as closely as we like, but we never 

exactly reach its value. For example, is 

' 25 
g 

just less than 2, and - is greater than 2, so 
4 

7 3 

that \/2 lies between -= and 5 . But the best 

o 1. 

systematic way of approximating to \/2 in 
obtaining a series of decimal fractions, each 
bigger than the last, is by the ordinary method 
of extracting the square root ; thus the series 

14 141 1414 

18 * 16' iob' looo' and so on ' 

Ratios of this sort are called by the Greeks 
incommensurable. They have excited from 
the time of the Greeks onwards a great deal 
of philosophic discussion, and the difficulties 
connected with them have only recently been 
cleared up. 

We will put the incommensurable ratios 
with the fractions, and consider the whole 
set of integral numbers, fractional numbers, 
and incommensurable numbers as forming 
one class of numbers which we will call " real 
numbers." We always think of the real 
numbers as arranged in order of magnitude, 
starting from zero and going upwards, and 
becoming indefinitely larger and larger as we 
proceed. The real numbers are conveniently 



represented by points on a line. Let OX be 






I 

2 


1 


3 
2 


2 


5 
2 


3 


7 
2 


4 




o 


M 


A 


N 


B 


P 


C 


Q 


D 


X 



any line bounded at O and stretching away in- 
definitely in the direction OX. Take any con- 
venieiit point, A, on it, so that OA represents 
the unit length ; and divide off lengths AB, 
BC, CD, and so on, each equal to OA. Then 
the point O represents the number 0, A the 
number 1, B the number 2, and so on. In 
fact the number represented by any point is 
the measure of its distance from O, in terms 
of the unit length OA. The points between 
O and A represent the proper fractions and 
the incommensurable numbers less than 1 ; 

the middle point of OA represents -, that of 

2 

o e 

AB represents -, that of BC represents -, and 

a a 

so on. In this way every point on OX repre- 
sents some one real number, and every real 
number is represented by some one point on 
OX. 

The series (or row) of points along OX, 
starting from O and moving regularly in the 
direction from to X, represents the real 
numbers as arranged in an ascending order 



GENERALIZATIONS OF NUMBERS 75 

of size, starting from zero and continually 
increasing as we go on. 

All this seems simple enough, but even at 
this stage there are some interesting ideas to 
be got at by dwelling on these obvious facts. 
Consider the series of points which represent 
the integral numbers only, namely, the points, 
O, A, B, C, D, etc. Here there is a first point 
0, a definite next point, A, and each point, 
such as A or B, has one definite immediate 
predecessor and one definite immediate suc- 
cessor, with the exception of 0, which has no 
predecessor ; also the series goes on in- 
definitely without end. This sort of order is 
called the type of order of the integers ; its 
essence is the possession of next-door neigh- 
bours on either side with the exception of 
No. 1 in the row. Again consider the integers 
and fractions together, omitting the points 
which correspond to the incommensurable 
ratios. The sort of serial order which we now 
obtain is quite different. There is a first 
term ; but no term has any immediate pre- 
decessor or immediate successor. This is 
easily seen to be the case, for between any 
two fractions we can always find another 
fraction intermediate in value. One very 
simple way of doing this is to add the fractions 
together and to halve the result. For ex- 
ample, between f and , the fraction (f + f), 
that is IT, lies ; and between f and H the 



76 INTRODUCTION TO MATHEMATICS 

fraction (f -j- {), that is If, lies ; and so on 
indefinitely. Because of this property the 
series is said to be " compact." There is no 
end point to the series, which increases in- 
definitely without limit as we go along the 
line OX. It would seem at first sight as 
though the type of series got in this way from 
the fractions, always including the integers, 
would be the same as that got from all the 
real numbers, integers, fractions, and incom- 
mensurables taken together, that is, from all 
the points on the line OX. All that we have 
hitherto said about the series of fractions 
applies equally well to the series of all real 
numbers. But there are important differ- 
ences which we now proceed to develop. The 
absence of the incommensurables from the 
series of fractions leaves an absence of end- 
points to certain classes. Thus, consider the 
incommensurable A/2. In the series of real 
numbers this stands between all the numbers 
whose squares are less than 2, and all the 
numbers whose squares are greater than 2. 
But keeping to the series of fractions alone 
and not thinking of the incommensurables, so 
that we cannot bring in \/2, there is no frac- 
tion which has the property of dividing off 
the series into two parts in this way, i.e. so 
that all the members on one side have their 
squares less than 2, and on the other side 
greater than 2. Hence in the series of frac- 



GENERALIZATIONS OF NUMBERS 77 

tions there is a quasi-gap where \/2 ought to 
come. This presence of quasi-gaps in the 
series of fractions may seem a small matter ; 
but any mathematician, who happens to read 
this, knows that the possible absence of limits 
or maxima to a class of numbers, which yet 
does not spread over the whole series of num- 
bers, is no small evil. It is to avoid this 
difficulty that recourse is had to the incom- 
mensurables, so as to obtain a complete series 
with no gaps. 

There is another even more fundamental 
difference between the two series. We can 
rearrange the fractions in a series like that of 
the integers, that is, with a first term, and 
such that each term has an immediate suc- 
cessor and (except the first term) an immediate 
predecessor. We can show how this can be 
done. Let every term in the series of fractions 
and integers be written in the fractional form 
by writing j for 1, f for 2, and so on for all the 
integers, excluding 0. Also for the moment 
we will reckon fractions which are equal in 
value but not reduced to their lowest terms 
as distinct ; so that, for example, until further 
notice f, |, J-, A> etc., are all reckoned as dis- 
tinct. Now group the fractions into classes 
by adding together the numerator and de- 
nominator of each term. For the sake of 
brevity call this sum of the numerator and 
denominator of a fraction its index. Thus 7 



78 INTRODUCTION TO MATHEMATICS 

is the index of |, and also of f , and of f . Let 
the fractions in each class be all fractions 
which have some specified index, which may 
therefore also be called the class index. Now 
arrange these classes in the order of magni- 
tude of their indices. The first class has 
the index 2, and its only member is T 5 the 
second class has the index 3, and its members 
are and f ; the third class has the index 
4, and its members are J, f, f; the fourth 
class has the index 5, and its members are 
i i> f T J an d so on - It is easy to see that 
the number of members (still including frac- 
tions not in their lowest terms) belonging to 
any class is one less than its index. Also the 
members of any one class can be arranged 
in order by taking the first member to be the 
fraction with numerator 1, the second mem- 
ber to have the numerator 2, and so on, up to 
(n 1) where n is the index. Thus for the 
class of index n, the members appear in the 
order. 

123 n-1 . 

~, -, -, . . ., = . The mem- 

n 1 n 2 n3 1 

bers of the first four classes have in fact been 
mentioned in this order. Thus the whole set 
of fractions have now been arranged in an 
order like that of the integers. It runs thus 

1121 T21 31234 

r 2' r ' LLl' r ? 3* 2' r 



GENERALIZATIONS OF NUMBERS 79 

n-2 _1 2_ 3 n-1 1 

1 'ra-l'n-2'n-3' ' '' 1 ' n' 

and so on. 

Now we can get rid of all repetitions of 
fractions of the same value by simply striking 
them out whenever they appear after their 
first occurrence. In the few initial terms 
written down above, f which is enclosed above 
in square brackets is the only fraction not in 
its lowest terms. It has occurred before as 
T. Thus this must be struck out. But the 
series is still left with the same properties, 
namely, (a) there is a first term, (6) each term 
has next-door neighbours, (c) the series goes 
on without end. 

It can be proved that it is not possible to 
arrange the whole series of real numbers in 
this way. This curious fact was discovered 
by Georg Cantor, a German mathematician 
still living ; it is of the utmost importance 
in the philosophy of mathematical ideas. We 
are here in fact touching on the fringe of the 
great problems of the meaning of continuity 
and of infinity. 

Another extension of number comes from 
the introduction of the idea of what has been 
variously named an operation or a step, 
names which are respectively appropriate 
from slightly different points of view. We 
will start with a particular case. Consider 



80 INTRODUCTION TO MATHEMATICS 

the statement 2+3=5. We add 3 to 2 and 
obtain 5. Think of the operation of adding 
3: let this be denoted by +3. Again 43 
=1. Think of the operation of subtracting 
3 : let this be denoted by 3. Thus instead 
of considering the real numbers in themselves, 
we consider the operations of adding or sub- 
tracting them : instead of -v/2, we consider 
+ V2 and \/2, namely the operations of 
adding V2 and of subtracting \/2. Then we 
can add these operations, of course in a 
different sense of addition to that in which we 
add numbers. The sum of two operations is 
the single operation which has the same effect 
as the two operations applied successively. 
In what order are the two operations to be 
applied ? The answer is that it is indifferent, 
since for example 

2+3+1=2+1+3; 

so that the addition of the steps +3 and +1 
is commutative. 

Mathematicians have a habit, which is 
puzzling to those engaged in tracing out 
meanings, but is very convenient in practice, 
of using the same symbol in different though 
allied senses. The one essential requisite for 
a symbol in their eyes is that, whatever its 
possible varieties of meaning, the formal laws 
for its use shall always be the same. In 



GENERALIZATIONS OF NUMBERS 81 

accordance with this habit the addition of 
operations is denoted by -f as well as the 
addition of numbers. Accordingly we can 
write 



where the middle + on the left-hand side 
denotes the addition of the operations -{-3 
and -f 1. But, furthermore, we need not be 
so very pedantic in our symbolism, except in 
the rare instances when we are directly tracing 
meanings ; thus we always drop the first + 
of a line and the brackets, and never write 
two + signs running. So the above equation 
becomes 

3+1=4, 

which we interpret as simple numerical addi- 
tion, or as the more elaborate addition of 
operations which is fully expressed in the 
previous way of writing the equation, or 
lastly as expressing the result of applying 
the operation +1 to the number 3 and ob- 
taining the number 4. Any interpretation 
which is possible is always correct. But the 
only interpretation which is always possible, 
under certain conditions, is that of operations. 
The other interpretations often give non- 
sensical results. 

This leads us at once to a question, which 
must have been rising insistently in the 



reader's mind : What is the use of all this 
elaboration ? At this point our friend, the 
practical man, will surely step in and insist on 
sweeping away all these silly cobwebs of the 
brain. The answer is that what the mathe- 
matician is seeking is Generality. This is an 
idea worthy to be placed beside the notions 
of the Variable and of Form so far as concerns 
its importance in governing mathematical 
procedure. Any limitation whatsoever upon 
the generality of theorems, or of proofs, or of 
interpretation is abhorrent to the mathe- 
matical instinct. These three notions, of the 
variable, of form, and of generality, compose 
a sort of mathematical trinity which preside 
over the whole subject. They all really 
spring from the same root, namely from the 
abstract nature of the science. 

Let us see how generality is gained by the 
introduction of this idea of operations. Take 
the equation x +1=3; the solution is x =2. 
Here we can interpret our symbols as mere 
numbers, and the recourse to " operations " 
is entirely unnecessary. But, if a? is a mere 
number, the equation #+3=1 is nonsense. 
For x should be the number of things which 
remain when you have taken 3 things away 
from 1 thing ; and no such procedure is 
possible. At this point our idea of algebraic 
form steps in, itself only generalization under 
another aspect. We consider, therefore, the 



GENERALIZATIONS OF NUMBERS 83 

general equation of the same form as <r+l =3, 
This equation is x-\-a=b y and its solution is 
x =b a. Here our difficulties become acute ; 
for this form can only be used for the numeri- 
cal interpretation so long as b is greater than 
a, and we cannot say without qualification 
that a and b may be any constants. In other 
words we have introduced a limitation on 
the variability of the " constants " a and b, 
which we must drag like a chain throughout 
all our reasoning. Really prolonged mathe- 
matical investigations would be impossible 
under such conditions. Every equation 
would at last be buried under a pile of limita- 
tions. But if we now interpret our symbols 
as " operations," all limitation vanishes like 
magic. The equation x +1=3 gives # = +2, 
the equation x +3=1 gives x= 2, the equa- 
tion x-\~a=b gives x=ba which is an opera- 
tion of addition or subtraction as the case 
may be. We need never decide whether b a 
represents the operation of addition or of 
subtraction, for the rules of procedure with 
the symbols are the same in either case. 

It does not fall within the plan of this work 1 
to write a detailed chapter of elementary 
algebra. Our object is merely to make plain 
the fundamental ideas which guide the forma- 
tion of the science. Accordingly we do not 
further explain the detailed rules by which 
the " positive and negative numbers " are 



84 INTRODUCTION TO MATHEMATICS 

multiplied and otherwise combined. We have 
explained above that positive and negative 
numbers are operations. They have also 
been called " steps." Thus +3 is the step 
by which we go from 2 to 5, and 3 is the 
step backwards by which we go from 5 to 2. 
Consider the line OX divided in the way ex- 
plained in the earlier part of the chapter, so 
that its points represent numbers. Then +2 

, D' C' B' A' +1 +2 +3 

-3-2-1 A B C D E 

is the step from to B, or from A to C, or 
(if the divisions are taken backwards along 
OX') from C' to A', or from D f to B', and so 
on. Similarly 2 is the step from to B', 
or from B' to D', or from B to 0, or from C 
to A. 

We may consider the point which is reached 
by a step from 0, as representative of that 
step. Thus A represents +1> B represents 
+2, A' represents 1, B' represents 2, and 
so on. It will be noted that, whereas previ- 
ously with the mere "unsigned " real numbers 
the points on one side of only, namely along 
OX, were representative of numbers, now 
with steps every point on the whole line 
stretching on both sides of is representative 
of a step. This is a pictorial representation 
of the superior generality introduced by the 
positive and negative numbers, namely the 



GENERALIZATIONS OF NUMBERS 85 

operations or steps. These "signed " num- 
bers are also particular cases of what have 
been called vectors (from the Latin veho, I 
draw or carry). For we may think of a 
particle as carried from O to A, or from A 
to B. 

In suggesting a few pages ago that the 
practical man would object to the subtlety 
involved by the introduction of the positive 
and negative numbers, we were libelling that 
excellent individual. For in truth we are on 
the scene of one of his greatest triumphs. If 
the truth must be confessed, it was the practi- 
cal man himself who first employed the actual 
symbols -f- and . Their origin is not very 
certain, but it seems most probable that they 
arose from the marks chalked on chests of 
goods in German warehouses, to denote excess 
or defect from some standard weight. The 
earliest notice of them occurs in a book pub- 
lished at Leipzig, in A.D. 1489. They seem 
first to have been employed in mathematics 
by a German mathematician, Stifel, in a book 
published at Nuremburg in 1544 A.D. But 
then it is only recently that the Germans 
have come to be looked on as emphatically 
a practical nation. There is an old epigram 
which assigns the empire of the sea to the 
English, of the land to the French, and of the 
clouds to the Germans. Surely it was from 
the clouds that the Germans fetched + and 



86 INTRODUCTION TO MATHEMATICS 

; the ideas which these symbols have 
generated are much too important for the 
welfare of humanity to have come from the 
sea or from the land. 

The possibilities of application of the posi- 
tive and negative numbers are very obvious. 
If lengths in one direction are represented 
by positive numbers, those in the opposite 
direction are represented by negative numbers. 
If a velocity in one direction is positive, that 
in the opposite direction is negative. If a 
rotation round a dial in the opposite direction 
to the hands of a clock (anti-clockwise) is 
positive, that in the clockwise direction is 
negative. If a balance at the bank is posi- 
tive, an overdraft is negative. If vitreous 
electrification is positive, resinous electrifica- 
tion is negative. Indeed, in this latter case, 
the terms positive electrification and negative 
electrification, considered as mere names, 
have practically driven out the other terms. 
An endless series of examples could be given. 
The idea of positive and negative numbers 
has been practically the most successful of 
mathematical subtleties. 



CHAPTER VII 

IMAGINARY NUMBERS 

IF the mathematical ideas dealt with in the 
last chapter have been a popular success, 
those of the present chapter have excited 
almost as much general attention. But their 
success has been of a different character, it 
has been what the French term a succes de 
scandale. Not only the practical man, but 
also men of letters and philosophers have ex- 
pressed their bewilderment at the devotion 
of mathematicians to mysterious entities 
which by their very name are confessed to be 
imaginary. At this point it may be useful 
to observe that a certain type of intellect 
is always worrying itself and others by 
discussion as to the applicability of technical 
terms. Are the incommensurable numbers 
properly called numbers ? Are the positive 
and negative numbers really numbers ? Are 
the imaginary numbers imaginary, and are 
they numbers ? are types of such futile 
questions. Now, it cannot be too clearly 
understood that, in science, technical terms 
are names arbitrarily assigned, like Christian 

87 



88 INTRODUCTION TO MATHEMATICS 

names to children. There can be no question 
of the names being right or wrong. They 
may be judicious or injudicious ; for they can 
sometimes be so arranged as to be easy to 
remember, or so as to suggest relevant and 
important ideas. But the essential principle 
involved was quite clearly enunciated in 
Wonderland to Alice by Humpty Dumpty, 
when he told her, a propos of his use of words, 
" I pay them extra and make them mean 
what I like." So we will not bother as to 
whether imaginary numbers are imaginary, 
or as to whether they are numbers, but will 
take the phrase as the arbitrary name of a 
certain mathematical idea, which we will now 
endeavour to make plain. 

The origin of the conception is in every 
way similar to that of the positive and nega- 
tive numbers. In exactly the same way it 
is due to the three great mathematical ideas 
of the variable, of algebraic form, and of 
generalization. The positive and negative 
numbers arose from the consideration of 
equations like #-fl=3, +3=l, and the 
general form x+a=b. Similarly the origin 
of imaginary numbers is due to equations like 
# 2 +l=8, a? 2 +3=1, and OJ 2 -fa=&. Exactly 
the same process is gone through. The equa- 
tion a; 2 +1 =3 becomes # 2 =2, and this has two 
solutions, either x = + V%, or x = V2. The 
statement that there are these alternative 



IMAGINARY NUMBERS 89 

solutions is usually written x = A/2. So far 
all is plain sailing, as it was in the previous 
case. But now an analogous difficulty arises. 
For the equation x 2 +3=1 gives # 2 = 2 and 
there is no positive or negative number which, 
when multiplied by itself, will give a negative 
square. Hence, if our symbols are to mean 
the ordinary positive or negative numbers, 
there is no solution to x 2 = 2, and the equa- 
tion is in fact nonsense. Thus, finally taking 
the general form x 2 +a=b ) we find the pair 
of solutions x = \/(b a), when, and only 
when, b is not less than a. Accordingly we 
cannot say unrestrictedly that the " con- 
stants " a and b may be any numbers, that is, 
the " constants " a and b are not, as they 
ought to be, independent unrestricted " vari- 
ables " ; and so again a host of limitations 
and restrictions will accumulate round our 
work as we proc'eed. 

The same task as before therefore awaits 
us : we must give a new interpretation to our 
symbols, so that the solutions V(6 a) for 
the equation x 2 -{-a=b always have meaning. 
In other words, we require an interpretation 
of the symbols so that Va always has meaning 
whether a be positive or negative. Of 
course, the interpretation must be such that 
all the ordinary formal laws for addition, sub- 
traction, multiplication, and division hold 
good ; and also it must not interfere with the 



90 INTRODUCTION TO MATHEMATICS 

generality which we have attained by the use 
of the positive and negative numbers. In 
fact, it must in a sense include them as 
special cases. When a is negative we may 
write c 2 for it, so that c 2 is positive. Then 



Va = V(^?) =y r ((->l)xc 2 } 

=V(-i) Vc*=c vT^i)- 

Hence, if we can so interpret our symbols that 
V( 1) has a meaning, we have attained our 
object. Thus V( 1) has come to be looked 
on as the head and forefront of all the 
imaginary quantities. 

This business of finding an interpretation 
for V( 1) is a much tougher job than the 
analogous one of interpreting 1. In fact, 
while the easier problem was solved almost 
instinctively as soon as it arose, it at first 
hardly occurred, even to the greatest mathe- 
maticians, that here a problem existed which 
was perhaps capable of solution. Equations 
like x 2 - 3, when they arose, were simply 
ruled aside as nonsense. 

However, it came to be gradually perceived 
during the eighteenth century, and even 
earlier, how very convenient it would be if 
an interpretation could be assigned to these 
nonsensical symbols. Formal reasoning with 
these symbols was gone through, merely 
assuming that they obeyed the ordinary 



IMAGINARY NUMBERS 91 

algebraic laws of transformation ; and it was 
seen that a whole world of interesting results 
could be attained, if only these symbols might 
legitimately be used. Many mathematicians 
were not then very clear as to the logic of 
their procedure, and an idea gained ground 
that, in some mysterious way, symbols which 
mean nothing can by appropriate manipula- 
tion yield valid proofs of propositions. No- 
thing can be more mistaken. A symbol 
which has not been properly defined is not a 
symbol at all. It is merely a blot of ink on 
paper which has an easily recognized shape. 
Nothing can be proved by a succession of 
blots, except the existence of a bad pen or a 
careless writer. It was during this epoch 
that the epithet "imaginary" came to be 
applied to V( 1). What these mathema- 
ticians had really succeeded in proving were 
a series of hypothetical propositions, of which 
this is the blank form: If interpretations 
exist for V( 1) and for the addition, sub- 
traction, multiplication, and division of 
V( 1) which make the ordinary algebraic 
rules (e.g. x+y=y-\-x, etc.) to be satisfied, 
then such and such results follows. It was 
natural that the mathematicians should not 
always appreciate the big "If," which ought 
to have preceded the statements of their re- 
suits. 

As may be expected the interpretation, 



92 INTRODUCTION TO MATHEMATICS 

when found, was a much more elaborate affair 
than that of the negative numbers and the 
reader's attention must be asked for some 
careful preliminary explanation. We have 
already come across the representation of a 
point by two numbers. By the aid of the 



p' 



M 1 



p" 



y' 

A'' 



y' 



ptu 



y 

Fig. 8. 

positive and negative numbers we can now 
represent the position of any point in a plane 
by a pair of such numbers. Thus we take 
the pair of straight lines XOX' and YOY' t at 
right angles, as the " axes " from which we 
start all our measurements. Lengths mea- 
sured along OX and OY are positive, and 
measured backwards along OX' and OY' are 
negative. Suppose that a pair of numbers, 
written in order,e.g. (+3, +!) so that there 



IMAGINARY NUMBERS 93 

is a first number (+3 in the above example), 
and a second number (+1 in the above ex- 
ample), represents measurements from O 
along XOX' for the first number, and along 
YOY' for the second number. Thus (cf . fig. 9) in 
( +3, +1) a length of 3 units is to be measured 
along XOX' in the positive direction, that 
is from O towards X, and a length +1 
measured along YOY' in the positive direc~ 
tion, that is from O towards F. Similarly in 
(3, +1) the length of 3 units is to be 
measured from O towards X', and of 1 unit 
from towards Y . Also in (3, 1) the 
two lengths are to be measured along OX' 
and OF' respectively, and in (+3, 1) along 
OX and OF* respectively. Let us for the 
moment call such a pair of numbers an 
" ordered couple." Then, from the two num- 
bers 1 and 3, eight ordered couples can be 
generated, namely 

(+1, +3), (-1, +3), (-1, -3), (+1, -3), 
(+3, +1), (-3, +1), (-3, -1), (+3, -1). 

Each of these eight "ordered couples " directs 
a process of measurement along XOX' and 
FOF' which is different from that directed 
by any of the others. 

The processes of measurement represented 
by the last four ordered couples, mentioned 
above, are given pictorially in the figure. 
The lengths OM and ON together correspond 



94 INTRODUCTION TO MATHEMATICS 

to (+3, +1), the lengths OM' and ON 
together correspond to (3, +1), OM' and 
ON' together to (3, 1), and OM and 
ON' together to (+3, 1). But by com- 
pleting the various rectangles, it is easy to 
see that the point P completely determines 
and is determined by the ordered couple 



P' 


N P 






fl 




. 


-3 


*3 


M 


H' 





-' 





P" 



N' 



y.' 
Fig. 9. 

(+3, +1), the point P' by (-3, -fl), the 
point P" by (-3, -1), and the point P'" by 
(+3, 1). More generally in the previous 
figure (8), the point P corresponds to the 
ordered couple (x, y), where x and y in the 
figure are both assumed to be positive, the 
point P' corresponds to (#', y), where x' in 
the figure is assumed to be negative, P" to 
(#' 2/')> and P'" to (x, y'). Thus an ordered 



IMAGINARY NUMBERS 95 

couple (x, y), where x and y are any positive 
or negative numbers, and the corresponding 
point reciprocally determine each other. It 
is convenient to introduce some names at this 
juncture. In the ordered couple (x, y) the 
first number x is called the " abscissa " of the 
corresponding point, and the second number 
y is called the " ordinate " of the point, and 
the two numbers together are called the " co- 
ordinates " of the point. The idea of deter- 
mining the position of a point by its " co- 
ordinates " was by no means new when the 
theory of " imaginaries " was being formed. 
It was due to Descartes, the great French 
mathematician and philosopher, and appears 
in his Discours published at Leyden in 1637 
A.D. The idea of the ordered couple as a 
thing on its own account is of later growth 
and is the outcome of the efforts to interpret 
imaginaries in the most abstract way possible. 

It may be noticed as a further illustration 
of this idea of the ordered couple, that the 
point M in fig. 9 is the couple (+3, 0), the 
point N is the couple (0, +1), the point M' 
the couple (3, 0), the point N' the couple 
(0, 1), the point O the couple (0, 0). 

Another way of representing the ordered 
couple (x, y) is to think of it as representing 
the dotted line OP (cf . fig. 8), rather than the 
point P. Thus the ordered couple represents 
a line drawn from an " origin," O, of a certain 



96 INTRODUCTION TO MATHEMATICS 

length and in a certain direction. The line 
OP may be called the vector line from O to 
P, or the step from O to P. We see, therefore, 
that we have in this chapter only extended 
the interpretation which we gave formerly of 
the positive and negative numbers. This 
method of representation by vectors is very 
useful when we consider the meaning to be 
assigned to the operations of the addition and 
multiplication of ordered couples. 

We will now go on to this question, and 
ask what meaning we shall find it convenient 
to assign to the addition of the two ordered 
couples (x, y) and (a?', y'). The interpreta- 
tion must, (a) make the result of addition 
to be another ordered couple, (b) make the 
operation commutative so that (x, y) + 
(x', y') =(%'Ty') -f-(tf71/)> (c) make the opera- 
tion associative so that 



(d) make the result of subtraction unique, 
so that when we seek to determine the 
unknown ordered couple (x, y) so as to 
satisfy the equation 

(x, y)+(a, b)=(c, d), 

there is one and only one answer which we 
can represent by 

(x, y)=(c, d)-(a t b). 



IMAGINARY NUMBERS 97 

All these requisites are satisfied by taking 
(x y y)+(x', y') to mean the ordered couple 
(x+x', y+y'). Accordingly by definition we 
put 

(*, t/)+(a?', y')=(x+x f , y+y'). 

Notice that here we have adopted the mathe- 
matical habit of using the same symbol -f- in 
different senses. The + on the left-hand side 
of the equation has the new meaning of + 
which we are just defining ; while the two 
-f-'s on the right-hand side have the meaning 
of the addition of positive and negative num- 
bers (operations) which was defined in the 
last chapter. No practical confusion arises 
from this double use. 

As examples of addition we have 

(+3, + l)+(+2, + 6) =(+5, + 7), 
(+3, - l)+(-2, - 6)=(+l - 7), 
( +8,+l)+(-8, -1)=(0, 0). 

The meaning of subtraction is now settled 
for us. We find that 

(x, y)(u, V)=(IK-U, yv). 
Thus 

{ +3, + 2) -( +1, + 1) =( +2, -f 1), 
and 

( +1, - 2) -( +2, - 4)=( -1, + 2), 
and 

( -1, - 2) -( +2, + 3) =( -3, - 5). 

D 



98 INTRODUCTION TO MATHEMATICS 

It is easy to see that 

(x, y)-(u, v)=(x, */)+(-, -v). 
Also 

0* y) (# y) = (> o). 

Hence (0, 0) is to be looked on as the zero 
ordered couple. For example 

( S/)-H 0)=(a?, y). 

The pictorial representation of the addition 
of ordered couples is surprisingly easy. 



y 


....-.i 


K 




4 
1 


r .-*7 








"' F 
^ 


/ 


S 





M, M H' 



Fig. 10. 

Let OP represent (x, y) so that OM=x 
and PM=y ; let OQ represent (x\, y\) so that 
OMi=Xi&nd QMi=t/i. Complete the paral- 
lelogram OPRQ by the dotted lines PR and 
QR, then the diagonal OR is the ordered 
couple (x-\-Xi, j/+t/i). For draw PS parallel 



IMAGINARY NUMBERS 99 

to OX ; then evidently the triangles OQM i 
and PRS are in all respects equal. Hence 
MM'=PS=Xi, and RS=QM l \ and there- 
fore 

OM'=OM+MM' = 
RM' =SM' +RS = 



Thus OJ? represents the ordered couple as 
required. This figure can also be drawn with 
OP and OQ in other quadrants. 

It is at once obvious that we have here 
come back to the parallelogram law, which 
was mentioned in Chapter VI. , on the laws of 
motion, as applying to velocities and forces. 
It will be remembered that, if OP and OQ 
represent two velocities, a particle is said to 
be moving with a velocity equal to the two 
velocities added together if it be moving with 
the velocity OR. In other words OR is said 
to be the resultant of the two velocities OP 
and OQ. Again forces acting at a point of a 
body can be represented by lines just as 
velocities can be ; and the same parallelogram 
law holds, namely, that the resultant of the 
two forces OP and OQ is the force represented 
by the diagonal OR. It follows that we can 
look on an ordered couple as representing a 
velocity or a force, and the rule which we 
have just given for the addition of ordered 
couples then represents the fundamental laws 
of mechanics for the addition of forces and 



100 INTRODUCTION TO MATHEMATICS 

velocities. One of the most fascinating 
characteristics of mathematics is the surpris- 
ing way in which the ideas and results of 
different parts of the subject dovetail into 
each other. During the discussions of this 
and the previous chapter we have been guided 
merely by the most abstract of pure mathe- 
matical considerations ; and yet at the end 
of them we have been led back to the most 
fundamental of all the laws of nature, laws 
which have to be in the mind of every engineer 
as he designs an engine, and of every naval 
architect as he calculates the stability of a 
ship. It is no paradox to say that in our 
most theoretical moods we may be nearest to 
our most practical applications* 



CHAPTER VIII 

IMAGINARY NUMBERS (Continued) 

THE definition of the multiplication of 
ordered couples is guided by exactly the same 
considerations as is that of their addition. 
The interpretation of multiplication must be 
such that 

(a) the result is another ordered couple, 
(/3) the operation is commutative, so that 

(x, y) x(o?', y')=(x', y'} x(#, t/), 
(7) the operation is associative, so that 

{(x, y)x(x f , y')} x (u, v) 
=(# J/)x{(' y')*(u, v)} 9 

(&) must make the result of division unique 
[with an exception for the case of the zero 
couple (0, 0)], so that when we seek to deter- 
mine the unknown couple (x, y) so as to 
satisfy the equation 

(x, y)x(a, b)=(c, d), 

there is one and only one answer, which we 
can represent by 

(*, S/)= (c, d)-H(a, 6), or by (, t/)= JA*> 
101 



102 INTRODUCTION TO MATHEMATICS 

(e) Furthermore the law involving both 
addition and multiplication, called the dis- 
tributive law, must be satisfied, namely 

(x,y)x{(a,b)+(c 9 d)} 
= {(*, y) x (a, 6)} +{(*, y) x (c, d)}. 

All these conditions (a), (), (7), (8), (e) can 
be satisfied by an interpretation which, 
though it looks complicated at first, is capable 
of a simple geometrical interpretation. 

By definition we put 

(x, y)x(x', y') = {(xx'-yy')> (xy' + x'y}} (A) 

This is the definition of the meaning of the 
symbol x when it is written between two 
ordered couples. It follows evidently from 
this definition that the result of multiplica- 
tion is another ordered couple, and that the 
value of the right-hand side of equation (A) 
is not altered by simultaneously interchanging 
x with x' t and y with y'. Hence conditions 
(a) and (/?) are evidently satisfied. The proof 
of the satisfaction of (7), (8), (e) is equally 
easy when we have given the geometrical 
interpretation, which we will proceed to do 
in a moment. But before doing this it will 
be interesting to pause and see whether we 
have attained the object for which all this 
elaboration was initiated. 

We came across equations of the form 
a? 2 = 3, to which no solutions could be 



IMAGINARY NUMBERS 103 

assigned in terms of positive and negative real 
numbers. We then found that all our diffi- 
culties would vanish if we could interpret the 
equation x 2 = 1, i.e., if we could so define 
V(-l) that V( 1) x V( 1)= 1. 

Now let us consider the three special 
ordered couples * (0,0), (1,0), and (0,1). 

We have already proved that 

(*, */)+(<>, 0)=(a, y). 
Furthermore we now have 

(x, t/)x(0, 0)=(0, 0). 

Hence both for addition and for multiplica- 
tion the couple (0,0) plays the part of zero in 
elementary arithmetic and algebra ; com- 
pare the above equations with c+0=a?, and 
x x 0=0. 

Again consider (1, 0) : this plays the part 
of 1 in elementary arithmetic and algebra. 
In these elementary sciences the special 
characteristic of 1 is that x xl=#, for all 
values of x. Now by our law of multiplica- 
tion 

(x t y) x (1, 0)= {(x- 0), (y +0)} = (*, y). 
Thus (1, 0) is the unit couple. 

* For the future we follow the custom of omitting the 
-f sign wherever possible, thus (1,0) stands for ( + 1,0) 
and (0,1) for (0, 



104 INTRODUCTION TO MATHEMATICS 

Finally consider (0,1) : this will interpret 
for us the symbol V( 1). The symbol must 
therefore possess the characteristic property 

that V( 1) x V( Z T)= 1. Now by the 
law of multiplication for ordered couples 

(0,1) x (0,1) = {(0 - 1), (0 + 0)} = ( -1, 0). 

But (1,0) is the unit couple, and (1, 0) 
is the negative unit couple ; so that (0,1) has 
the desired property. There are, however, 
two roots of 1 to be provided for, namely 
V( 1). Consider (0, 1) ; here again re- 
membering that (1) 2 =1, we find, (0, 1) 
x(0,-l) = (-l, 0). 

Thus (0, 1) is the other square root of 
V( 1). Accordingly the ordered couples 
(0,1) and (0, 1) are the interpretations of 
V( 1) in terms of ordered couples. But 
which corresponds to which ? Does (0,1) 
correspond to + V( 1) and (0, 1) to 



- VC^T), or (0,1) to -Vr^andfO, - 1) 
to + V( 1) ? The answer is that it is per- 
fectly indifferent which symbolism we adopt. 
The ordered couples can be divided into 
three types, (i) the " complex imaginary " 
type (x,y), in which neither x nor y is zero ; 
(ii) the " real " type (#,0) ; (iii) the " pure 
imaginary " type (O,?/). Let us consider the 
relations of these types to each other. First 
multiply together the " complex imaginary ' 



IMAGINARY NUMBERS 105 

eouple (x,y) and the " real " couple (a,0), we 
find 

(a,0)x(x,y)=(ax, ay). 

Thus the effect is merely to multiply each 
term of the couple (x,y) by the positive or 
negative real number a. 

Secondly, multiply together the " complex 
imaginary " couple (x,y) and the " pure 
imaginary " couple (0,6), we find 

(0,6) x(x,y)=(by, bx). 

Here the effect is more complicated, and is 
best comprehended in the geometrical inter- 
pretation to which we proceed after noting 
three yet more special cases. 

Thirdly, we multiply the " real " couple 
(o,0) by the imaginary (0,6) and obtain 

(a,0) x (0,6) =(0,ai). 

Fourthly, we multiply the two " real " 
couples (a,0) and (a', 0) and obtain 

(a,0)x(a',0)=(oa',0). 

Fifthly, we multiply the two "imaginary 
couples " (0,6) and (0, 6) and obtain 

(0,6)x(0,6')=(-66', 0). 

We now turn to the geometrical interpreta- 
tion, beginning first with some special cases. 



106 INTRODUCTION TO MATHEMATICS 

Take the couples (1,3) and (2,0) and consider 
the equation 

(2,0) x (1,3) =(2,6) 





n 

u>. 


/ 


if 


R 




N,/ 






^^^ 


X 1 


_^\ 


t 




X 


M, M N 




Fig. 11. 



In the diagram (fig. 11) the vector OP re- 
presents (1, 3), and the vector ON represents 
(2,0), and the vector OQ represents (2,6). 
Thus the product (2,0) x(l,3) is found geo- 
metrically by taking the length of the vector 
OQ to be the product of the lengths of the 
vectors OP and ON, and (in this case) by 
producing OP to Q to be of the required 
length. Again, consider the product (0,2) x 
(1,3), we have 

(0, 2)x(l, 3)=(-6, 2) 

The vector ONi, corresponds to (0, 2) and 
the vector OR to (-6,2). Thus OR which 



IMAGINARY NUMBERS 107 

represents the new product is at right angles 
to OQ and of the same length. Notice that 
we have the same law regulating the length 
of OQ as in the previous case, namely, that 
its length is the product of the lengths of 
the two vectors which are multiplied to- 
gether ; but now that we have ONi along the 
*' ordinate " axis OY, instead of ON along 
the ** abscissa " axis OX, the direction of 
OP has been turned through a right- angle. 

Hitherto in these examples of multiplication 
we have looked on the vector OP as modified 
by the vectors ON and ONi. We shall get 
a clue to the general law for the direction by 
inverting the way of thought, and by think- 
ing of the vectors ON and ONi as modified by 
the vector OP. The law for the length re- 
mains unaffected ; the resultant length is the 
length of the product of the two vectors. 
The new direction for the enlarged ON (i.e. 
OQ) is found by rotating it in the (anti-clock- 
wise) direction of rotation from OX towards 
OY through an angle equal to the angle XOP : 
it is an accident of this particular case that 
this rotation makes OQ lie along the line OP. 
Again consider the product of ONi and OP ; 
the new direction for the enlarged ONi (i.e. 
OR) is found by rotating ON in the anti- 
clockwise direction of rotation through an 
angle equal to the angle XOP, namely, the 
angle NiOR is equal to the angle XOP. 



108 INTRODUCTION TO MATHEMATICS 

The general rule for the geometrical repre- 
sentation of multiplication can now be enunci- 
ated thus : 




Fig. 12. 

The product of the two vectors OP and 
OQ is a vector OR, whose length is the pro- 
duct of the lengths of OP and OQ and whose 
direction OR is such that the angle XOR is 
equal to the sum of the angles XOP and XOQ. 

Hence we can conceive the vector OP as 
making the vector OQ rotate through an 
angle XOP (i.e. the angle QO.K = the angle 
XOP), or the vector OQ as making the vector 
OP rotate through the angle XOQ (i.e. the 
angle POR =the angle XOQ). 

We do not prove this general law, as we 



IMAGINARY NUMBERS 109 

should thereby be led into more technical 
processes of mathematics than falls within the 
design of this book. But now we can im- 
mediately see that the associative law [num- 
bered (7) above] for multiplication is satisfied. 
Consider first the length of the resultant 
vector ; this is got by the ordinary process 
of multiplication for real numbers ; and thus 
the associative law holds for it. 

Again, the direction of the resultant vector 
is got by the mere addition of angles, and the 
associative law holds for this process also. 

So much for multiplication. We have now 
rapidly indicated, by considering addition and 
multiplication, how an algebra or " calculus " 
of vectors in one plane can be constructed, 
which is such that any two vectors in the 
plane can be added, or subtracted, and can 
be multiplied, or divided one by the other. 

We have not considered the technical de- 
tails of all these processes because it would 
lead us too far into mathematical details ; 
but we have shown the general mode of pro- 
cedure. When we are interpreting our alge- 
braic symbols in this way, we are said to be 
employing " imaginary quantities " or *' com- 
plex quantities." These terms are mere 
details, and we have far too much to think 
about to stop to enquire whether they are or 
are not very happily chosen. 

The nett result of our investigations is that 



110 INTRODUCTION TO MATHEMATICS 

any equations like #+3=2 or (#+3) 2 = 2 
can now always be interpreted into terms of 
vectors, and solutions found for them. In 
seeking for such interpretations it is well to 
note that 3 becomes (3,0), and 2 becomes 
(2,0), and x becomes the "unknown" 
couple (u, v) : so the two equations become 
respectively (u, v) +(3,0) =(2,0), and {(u,v) 
+(3,0)}2 = (-2,0). 

We have now completely solved the initial 
difficulties which caught our eye as soon as 
we considered even the elements of algebra. 
The science as it emerges from the solution is 
much more complex in ideas than that with 
which we started. We have, in fact, created 
a new and entirely different science, which 
will serve all the purposes for which the old 
science was invented and many more in addi- 
tion. But, before we can congratulate our- 
selves on this result to our labours, we must 
allay a suspicion which ought by this tune to 
have arisen in the mind of the student. The 
question which the reader ought to be asking 
himself is : Where is all this invention of new 
interpretations going to end ? It is true that 
we have succeeded in interpreting algebra so 
as always to be able to solve a quadratic 
equation like a? 2 2#+4=0; but there are 
an endless number of other equations, for 
example, x 3 2a?-f4=0, # 4 +a? 3 +2=0, and so 
on without limit. Have we got to make a 



IMAGINARY NUMBERS 111 

new science whenever a new equation ap- 
pears ? 

Now, if this were the case, the whole of our 
preceding investigations, though to some 
minds they might be amusing, would in truth 
be of very trifling importance. But the great 
fact, which has made modern analysis possible, 
is that, by the aid of this calculus of vectors, 
every formula which arises can receive its 
proper interpretation ; and the " unknown " 
quantity in every equation can be shown to 
indicate some vector. Thus the science is now 
complete in itself as far as its fundamental 
ideas are concerned. It was receiving its final 
form about the same time as when the steam 
engine was being perfected, and will remain 
a great and powerful weapon for the achieve- 
ment of the victory of thought over things 
when curious specimens of that machine 
repose in museums in company with the 
helmets and breastplates of a slightly earlier 
epoch. 



CHAPTER IX 

COORDINATE GEOMETRY 

THE methods and ideas of coordinate geo- 
metry have already been employed in the 
previous chapters. It is now time for us to 
consider them more closely for their own 
sake ; and in doing so we shall strengthen our 
hold on other ideas to which we have attained. 
In the present and succeeding chapters we 
will go back to the idea of the positive and 
negative real numbers and will ignore the 
imaginaries which were introduced in the last 
two chapters. 

We have been perpetually using the idea 
that, by taking two axes, XOX' and YOY', 
in a plane, any point P in that plane can be 
determined in position by a pair of positive 
or negative numbers x and t/, where (cf. 
fig. 13) x is the length OM and y is the length 
PM. This conception, simple as it looks, is 
the main idea of the great subject of co- 
ordinate geometry. Its discovery marks a 
momentous epoch in the history of mathe- 
matical thought. It is due (as has been 
112 



COORDINATE GEOMETRY 113 



already said) to the philosopher Descartes, 
and occurred to him as an important mathe- 
matical method one morning as he lay in bed. 
Philosophers, when they have possessed a 
thorough knowledge of mathematics, have 
been among those who have enriched the 



p 

\y 



M 



Y' 

Fig. 13. 

science with some of its best ideas. On the 
other hand it must be said that, with hardly 
an exception, all the remarks on mathematics 
made by those philosophers who have pos- 
sessed but a slight or hasty and late-acquired 
knowledge of it are entirely worthless, being 
either trivial or wrong. The fact is a curious 
one ; since the ultimate ideas of mathematics 



114 INTRODUCTION TO MATHEMATICS 

seem, after all, to be very simple, almost 
childishly so, and to lie well within the 
province of philosophical thought. Probably 
their very simplicity is the cause of error ; we 
are not used to think about such simple 
abstract things, and a long training is neces- 
sary to secure even a partial immunity from 
error as soon as we diverge from the beaten 
track of thought. 

The discovery of coordinate geometry, and 
also that of projective geometry about the 
same time, illustrate another fact which is 
being continually verified in the history of 
knowledge, namely, that some of the greatest 
discoveries are to be made among the most 
well-known topics. By the time that the 
seventeenth century had arrived, geometry 
had already been studied for over two thousand 
years, even if we date its rise with the Greeks. 
Euclid, taught in the University of Alexandria, 
being born about 330 B.C. ; and he only 
systematized and extended the work of a long 
series of predecessors, some of them men of 
genius. After him generation after genera- 
tion of mathematicians laboured at the im- 
provement of the subject. Nor did the 
subject suffer from that fatal bar to progress, 
namely, that its study was confined to a 
narrow group of men of similar origin and 
outlook quite the contrary was the case ; 
by the seventeenth century it had passed 



COORDINATE GEOMETRY 115 

through the minds of Egyptians and Greeks, 
of Arabs and of Germans. And yet, after all 
this labour devoted to it through so many 
ages by such diverse minds its most important 
secrets were yet to be discovered. 

No one can have studied even the elements 
of elementary geometry without feeling the 
lack of some guiding method. Every proposi- 
tion has to be proved by a fresh display of in- 
genuity ; and a science for which this is true 
lacks the great requisite of scientific thought, 
namely, method. Now the especial point of 
coordinate geometry is that for the first 
time it introduced method. The remote 
deductions of a mathematical science are not 
of primary theoretical importance. The 
science has not been perfected, until it consists 
in essence of the exhibition of great allied 
methods by which information, on any desired 
topic which falls within its scope, can easily 
be obtained. The growth of a science is not 
primarily in bulk, but in ideas ; and the more 
the ideas grow, the fewer are the deductions 
which it is worth while to write down. Un- 
fortunately, mathematics is always encum- 
bered by the repetition in text-books of 
numberless subsidiary propositions, whose im- 
portance has been lost by their absorption 
into the role of particular cases of more 
general truths and, as we have already in- 
sisted, generality is the soul of mathematics. 



116 INTRODUCTION TO MATHEMATICS 

Again, coordinate geometry illustrates 
another feature of mathematics which has 
already been pointed out, namely, that mathe- 
matical sciences as they develop dovetail into 
each other, and share the same ideas in com- 
mon. It is not too much to say that the 
various branches of mathematics undergo a 
perpetual process of generalization, and that 
as they become generalized, they coalesce. 
Here again the reason springs from the very 
nature of the science, its generality, that is 
to say, from the fact that the science deals 
with the general truths which apply to all 
things in virtue of their very existence as 
things. In this connection the interest of co- 
ordinate geometry lies in the fact that it 
relates together geometry, which started as 
the science of space, and algebra, which has 
its origin in the science of number. 

Let us now recall the main ideas of the two 
sciences, and then see how they are related 
by Descartes' method of coordinates. Take 
algebra in the first place. We will not trouble 
ourselves about the imaginaries and will 
think merely of the real numbers with posi- 
tive or negative signs. The fundamental idea 
is that of any number, the variable number, 
which is denoted by a letter and not by any 
definite numeral. We then proceed to the 
consideration of correlations between vari- 
ables. For example, if x and y are two vari- 



ables, we may conceive them as correlated by 
the equations x+y=I, or by x y=I, or in 
any one of an indefinite number of other ways. 
This at once leads to the application of the 
idea of algebraic form. We think, in fact, of 
any correlation of some interesting type, thus 
rising from the initial conception of vari- 
able numbers to the secondary conception of 
variable correlations of numbers. Thus we 
generalize the correlation x+yl, into the 
correlation ax+by=c. Here a and b and c, 
being letters, stand for any numbers and are 
in fact themselves variables. But they are 
the variables which determine the variable 
correlation ; and the correlation, when deter- 
mined, correlates the variable numbers a: and 
y. Variables, like a, b, and c above, which 
are used to determine the correlation are 
called " constants," or parameters. The use 
of the term " constant " in this connection 
for what is really a variable may seem at first 
sight to be odd ; but it is really very natural. 
For the mathematical investigation is con- 
cerned with the relation between the variables 
x and y, after a,b,e are supposed to have been 
determined. So in a sense, relatively to x 
and y, the " constants " a, b, and c are con- 
stants. Thus ax+by=c stands for the general 
example of a certain algebraic form, that is, 
for a variable correlation belonging to a certain 
class. 



118 INTRODUCTION TO MATHEMATICS 



Again we generalize # 2 +t/ 2 =l into 
by 2 =c, or still further into ax 2 +2hxy +by 2 
=c, or, still further, into ax 2 +2hxy +by 2 +2gx 



Here again we are led to variable correlations 
which are indicated by their various algebraic 
forms. 

Now let us turn to geometry. The name 
of the science at once recalls to our minds 
the thought of figures and diagrams exhibiting 
triangles and rectangles and squares and 
circles, all in special relations to each other. 
The study of the simple properties of these 
figures is the subject matter of elementary 
geometry, as it is rightly presented to the 
beginner. Yet a moment's thought will show 
that this is not the true conception of the 
subject. It may be right for a child to com- 
mence his geometrical reasoning on shapes, 
like triangles and squares, which he has cut 
out with scissors. What, however, is a tri- 
angle ? It is a figure marked out and bounded 
by three bits of three straight lines. 

Now the boundary of spaces by bits of 
lines is a ve"ry complicated idea, and not at 
all one which gives any hope of exhibiting 
the simple general conceptions which should 
form the bones of the subject. We want 
something more simple and more general. It 
is this obsession with the wrong initial ideas 
very natural and good ideas for the creation 



COORDINATE GEOMETRY 119 

of first thoughts on the subject which was 
the cause of the comparative sterility of the 
study of the science during so many centuries. 
Coordinate geometry, and Descartes its in- 
ventor, must have the credit of disclosing the 
true simple objects for geometrical thought. 

In the place of a bit of a straight line, let 
us think of the whole of a straight line 
throughout its unending length in both direc- 
tions. This is the sort of general idea from 
which to start our geometrical investigations. 
The Greeks never seem to have found any 
use for this conception which is now funda- 
mental in all modern geometrical thought. 
Euclid always contemplates a straight line as 
drawn between two definite points, and is 
very careful to mention when it is to be pro- 
duced beyond this segment. He never thinks 
of the line as an entity given once for all as a 
whole. This careful definition and limita- 
tion, so as to exclude an infinity not immedi- 
ately apparent to the senses, was very charac- 
teristic of the Greeks in all their many 
activities. It is enshrined in the difference 
between Greek architecture and Gothic archi- 
tecture, and between the Greek religion and 
the modern religion. The spire on a Gothic 
cathedral and the importance of the un- 
bounded straight line in modern geometry 
are both emblematic of the transformation of 
the modern world. 



120 INTRODUCTION TO MATHEMATICS 

The straight line, considered as a whole, 
is accordingly the root idea from which 
modern geometry starts. But then other 
sorts of lines occur to us, and we arrive at the 
conception of the complete curve which at 
every point of it exhibits some uniform char- 
acteristic, just as the straight line exhibits 
at all points the characteristic of straight- 
ness. For example, there is the circle which 
at all points exhibits the characteristic of 
being at a given distance from its centre, and 
again there is the ellipse, which is an oval 
curve, such that the sum of the two distances 
of any point on it from two fixed points, called 
its foci, is constant for all points on the curve. 
It is evident that a circle is merely a particu- 
lar case of an ellipse when the two foci are 
superposed in the same point ; for then the 
sum of the two distances is merely twice the 
radius of the circle. The ancients knew the 
properties of the ellipse and the circle and, of 
course, considered them as wholes. For ex- 
ample, Euclid never starts with mere seg- 
ments (i.e., bits) of circles, which are then pro- 
longed. He always considers the whole circle 
as described. It is unfortunate that the 
circle is not the true fundamental line in 
geometry, so that his defective consideration 
of the straight line might have been of less 
consequence. 

This general idea of a curve which at any 



COORDINATE GEOMETRY 121 



point of it exhibits some uniform property is 
expressed in geometry by the term " locus." 
A locus is the curve (or surface, if we do not 
confine ourselves to a plane) formed by points, 
all of which possess some given property. 
To every property in relation to each other 
which points can have, there corresponds 
some locus, which consists of all the points 
possessing the property. In investigating 
the properties of a locus considered as a whole, 
we consider any point or points on the locus. 
Thus in geometry we again meet with the 
fundamental idea of the variable. Further- 
more, in classifying loci under such headings 
as straight lines, circles, ellipses, etc., we again 
find the idea of form. 

Accordingly, as in algebra we are concerned 
with variable numbers, correlations between 
variable numbers, and the classification of 
correlations into types by the idea of algebraic 
form ; so in geometry we are concerned with 
variable points, variable points satisfying 
some condition so as form to a locus, and the 
classification of loci into types by the idea of 
conditions of the same form. 

Now, the essence of coordinate geometry 
is the identification of the algebraic corre- 
lation with the geometrical locus. The point 
on a plane is represented in algebra by its 
two coordinates, x and t/, and the condition 
satisfied by any point on the locus Is re- 



122 INTRODUCTION TO MATHEMATICS 

presented by the corresponding correlation 
between x and y. Finally to correlations 
expressible in some general algebraic form, 
such as ax+by=c, there correspond loci of 
some general type, whose geometrical con- 
ditions are all of the same form. We 
have thus arrived at a position where we 
can effect a complete interchange in ideas 
and results between the two sciences. Each 
science throws light on the other, and itself 
gains immeasurably in power. It is im- 
possible not to feel stirred at the thought 
of the emotions of men at certain historic 
moments of adventure and discovery 
Columbus when he first saw the Western 
shore, Pizarro when he stared at the Pacific 
Ocean, Franklin when the electric spark came 
from the string of his kite, Galileo when he 
first turned his telescope to the heavens. 
Such moments are also granted to students 
in the abstract regions of thought, and high 
among them must be placed the morning when 
Descartes lay in bed and inventedfthe method 
of coordinate geometry. 

When one has once grasped the idea of co- 
ordinate geometry, the immediate question 
which starts to the mind is, What sort of 
loci correspond to the well-known algebraic 
forms ? For example, the simplest among 
the general types of algebraic forms is ax + 
by=c. The sort of locus which corresponds 



COORDINATE GEOMETRY 123 



to this is a straight line, and conversely to 
every straight line there corresponds an equa 
tion of this form. It is fortunate that the 
simplest among the geometrical loci should 
correspond to the simplest among the alge- 
braic forms. Indeed, it is this general corre- 
spondence of geometrical and algebraic sim- 
plicity which gives to the whole subject its 
power. It springs from the fact that the 
connection between geometry and algebra is 
not casual and artificial, but deep-seated and 
essential. The equation which corresponds 
to a locus is called the equation " of " (or 
" to ") the locus. Some examples of equations 
of straight lines will illustrate the subject. 



; 







Fig. 14. 



124 INTRODUCTION TO MATHEMATICS 

Consider y x =0 ; here the a, 6, and c, of 
the general form have been replaced by 1, 1, 
and respectively. This line passes through 
the "origin," 0, in the diagram and bisects 
the angle XOY. It is the line L'OL of the 
diagram. The fact that it passes through the 
origin, 0, is easily seen by observing that the 
equation is satisfied by putting xQ and 
t/=0 simultaneously, but and are the co- 
ordinates of 0. In fact it is easy to generalize 
and to see by the same method that the 
equation of any line through the origin is of 
the form ax-\-by=Q. The loeus of equation 
/+aj=0 also passes through the origin and 
bisects the angle X'OY : it is the line L\OL\ 
of the diagram. 

Consider y x=\ : the corresponding locus 
does not pass through the origin. We there- 
fore seek where it cuts the axes. It must cut 
the axis of x at some point of coordinates 
x and 0. But putting t/=0 in the equation, 
we get x 1; so the coordinates of this 
point (A) are 1 and 0. Similarly the point 
(B) where the line cuts the axis OY are and 
1. The locus is the line AB in the figure and 
is parallel to LOU. Similarly y+x=\ is the 
equation of line AJB of the figure ; and the 
locus is parallel to L^OL\. It is easy to prove 
the general theorem that two lines represented 
by equations of the forms ax-\-by=0 and 
ax+by=c are parallel. 



COORDINATE GEOMETRY 125 

The group of loci which we next come upon 
are sufficiently important to deserve a chap- 
ter to themselves. But before going on to 
them we will dwell a little longer on the main 
ideas of the subject. 

The position of any point P is determined 
by arbitrarily choosing an origin, 0, two axes, 
OX and OF, at right-angles, and then by 
noting its coordinates ac and y, i.e. OM and 
PM (cf. fig. 13). Also, as we have seen in the 
last chapter, P can be determined by the 
" vector " OP, where the idea of the vector 
includes a determinate direction as well as a 
determinate length. From an abstract 
mathematical point of view the idea of an 
arbitrary origin may appear artificial and 
clumsy, and similarly for the arbitrarily 
drawn axes, OX and OY. But in relation to 
the application of mathematics to the event 
of the Universe we are here symbolizing with 
direct simplicity the most fundamental fact 
respecting the outlook on the world afforded 
to us by our senses. We each of us refer 
our sensible perceptions of things to an origin 
which we call " here " : our location in a 
particular part of space round which we 
group the whole Universe is the essential fact 
of our bodily existence. We can imagine 
beings who observe all phenomena in all space 
with an equal eye, unbiassed in favour of any 
part. With us it is otherwise, a cat at our 



126 INTRODUCTION TO MATHEMATICS 

feet claims more attention than an earth- 
quake at Cape Horn, or than the destruction 
of a world in the Milky Way. It is true that 
in making a common stock of our knowledge 
with our fellowmen, we have to waive some- 
thing of the strict egoism of our own indi- 
vidual " here." We substitute " nearly 
here " for " here " ; thus we measure miles 
from the town hall of the nearest town, or 
from the capital of the country. In measur- 
ing the earth, men of science will put the 
origin at the earth's centre ; astronomers 
even rise to the extreme altruism of putting 
their origin inside the sun. But, far as this 
last origin may be, and even if we go further 
to some convenient point amid the nearer 
fixed stars, yet, compared to the immeasur- 
able infinities of space, it remains true that 
our first procedure in exploring the Universe 
is to fix upon an origin " nearly here." 

Again the relation of the coordinates OM 
and MP (i.e. x and t/) to the vector OP is an 
instance of the famous parallelogram law, as 
can easily be seen (cf. fig. 8) by completing 
the parallelogram OMPN. The idea of the 
" vector " OP, that is, of a directed magni- 
tude, is the root-idea of physical science. 
Any moving body has a certain magnitude 
of velocity in a certain direction, that is to 
say, its velocity is a directed magnitude, a 
vector. Again a force has a certain magni- 



COORDINATE GEOMETRY 127 

tude and has a definite direction. Thus, 
when in analytical geometry the ideas of the 
" origin," of " coordinates," and of " vec- 
tors " are introduced, we are studying the 
abstract conceptions which correspond to the 
fundamental facts of the physical world. 



CHAPTER X 

CONIC SECTIONS 

WHEN the Greek geometers had exhausted, 
as they thought, the more obvious and inter- 
esting properties of figures made up of 
straight lines and circles, they turned to 
the study of other curves ; and, with their 
almost infallible instinct for hitting upon 
things worth thinking about, they chiefly 

(devoted themselves to conic sections, that 
is, to the curves in which planes would cut 
the surfaces of circular cones. The man 
who must have the credit of inventing the 
study is Menaechmus (born 375 B.C. and 
died 325 B.C.); he was a pupil of Plato 
and one of the tutors of Alexander the 
Great. Alexander, by the by, is a con- 
spicuous example of the advantages of good 
tuition, for another of his tutors was the 
philosopher Aristotle. We may suspect that 
Alexander found Menaechmus rather a dull 
teacher, for it is related that he asked for the 

128 



CONIC SECTIONS 129 

proofs to be made shorter. It was to this 
request that Menaechmus replied : ** In the 
country there are private and even royal 
roads, but in geometry there is only one road 
for all." This reply no doubt was true 
enough in the sense in which it would have 
been immediately understood by Alexander. 
But if Menaechmus thought that his proofs 
could not be shortened, he was grievously 
mistaken ; and most modern mathematicians 
would be horribly bored, if they were com- 
pelled to study the Greek proofs of the pro- 
perties of conic sections. Nothing illustrates 
better the gain in power which is obtained by 
the introduction of relevant ideas into a 
science than to observe the progressive 
shortening of proofs which accompanies the 
growth of richness in idea. There is a cer- 
tain type of mathematician who is always 
rather impatient at delaying over the ideas 
of a subject : he is anxious at once to get on 
to the proofs of " important " problems. The 
history of the science is entirely against him. 
There are royal roads in science ; but those 
who first tread them are men of genius and 
not kings. 

The way in which conic sections first pre- 
sented themselves to mathematicians was as 
follows : think of a cone (cf. fig. 15), whose 
vertex (or point) is F, standing on a circular 
base STU, For example, a conical shade to 
E 



130 INTRODUCTION TO MATHEMATICS 

an electric light is often an example of such a 
surface. Now let the " generating " lines 
which pass through V and lie on the surface 
be all produced backwards; the result is a 
double cone, and PQR is another circular cross 
section on the opposite side of V to the cross 
section STU. The axis of the cone CVC' 
passes through all the centres of these circles 
and is perpendicular to their planes, which 
are parallel to each other. In the diagram 
the parts of the curves which are supposed 
to lie behind the plane of the paper are dotted 
lines, and the parts on the plane or in front 
of it are continuous lines. Now suppose this 
double cone is cut by a plane not perpen- 
dicular to the axis CVC', or at least not 
necessarily perpendicular to it. Then three 
cases can arise : 

(1) The plane may cut the cone in a closed 
oval curve, such as ABA'B' which lies en- 
tirely on one of the two half-cones. In this 
case the plane will not meet the other half-cone 
at all. Such a curve is called an ellipse ; it is 
an oval curve. A particular case of such a 
section of the cone is when the plane is per- 
pendicular to the axis CFC", then the section, 
such as STU or PQR, is a circle. IJe^nce a 
circle is a particular case of the ellipse. 

^^ ^ 

(2) The plane may be parallelled -tangent 
plane touching the cone along one of its " gen- 
erating " lines as for example the plane of the 



CONIC SECTIONS 131 

curve DiAiDi' in the diagram is parallel to 
the tangent plane touching the cone along the 
generating line VS ; the curve is still confined 
to one of the half-cones, but it is now not a 
closed oval curve, it goes on endlessly as long 
as the generating lines of the half-cone are 
produced away from the vertex. Such a 
conic section is called a<sarabola., / 

(3) The plane may cut botlFtTie half-cones, 
so that the complete curve consists of two 
detached portions, or " branches " as they 
are called, this case is illustrated by the two 
branches G^^G-i and LiA^L^ which together 
make up the curve. Neither branch is closed, 
each of them spreading out endlessly as the 
two half -cones are prolonged away from the 
vertex. Such a conic section is called a 
yperbolay 

There are accordingly three types of conic 
sections, namely, ellipses, parabolas, and 
hyperbolas. It is easy to see that, in a sense, 
parabolas are limiting cases lying between 
ellipses and hyperbolas. They form a more 
special sort and have to satisfy a more par- 
ticular condition. These three names are 
apparently due to Apollonius of Perga (born 
about 260 B.C., and died about 200 B.C.), who 
wrote a systematic treatise on conic sections 
which remained the standard work till the 
sixteenth century. 

It must at once be apparent how awkward 



and difficult the investigation of the proper- 
ties of these curves must have been to the 
Greek geometers. The curves are plane 
curves, and yet their investigation involves 




Fig. 16 

the drawing in perspective of a solid figure. 
Thus in the diagram given above we have 
practically drawn no subsidiary lines and yet 
the figure is sufficiently complicated. The 



CONIC SECTIONS 



133 



curves are plane curves, and it seems obvious 
that we should be able to define them without 




going beyond the plane into a solid figure. 
At the same time, just as in the " solid " 




Fig. 17 

definition there is one uniform method of 
definition namely, the section of a cone by 



134 INTRODUCTION TO MATHEMATICS 



a plane which yields three cases, so in any 
" plane " definition there also should be one 
uniform method of procedure which falls into 
three cases. Their shapes when drawn on 
their planes are those of the curved lines in 
the three figures 16, 17, and 18. The 
points A and A' in the figures are called 




Fig. 18 

the vertices and the line AA' the major axis. 
It will be noted that a parabola (cf. fig. 17) 
has only one vertex. Apollonius proved * that 

the ratio of PM 2 to AM.MA 



' (i.e. -^^- 
\ AM.MA 



remains constant both for the ellipse and the 
hyperbola (figs. 16 and 18), and that the ratio 

* Cf. Ball, loc. cit., for this account of Apollonius and 
Pappus. 



CONIC SECTIONS 135 

of PM 2 to AM is constant for the parabola 
of fig. 17 ; and he bases most of his work 
on this fact. We are evidently advancing 
towards the desired uniform definition which 
does not go out of the plane ; but have not 
yet quite attained to uniformity. 

In the diagrams 16 and 18, two points, S 
and S', will be seen marked, and in diagram 17 
one point, S. These are the foci of the curves, 
and are points of the greatest importance. 
Apollonius knew that for an ellipse the sum 
of SP and S'P (i.e. SP+S'P) is constant as 
P moves on the curve, and is equal to A A'. 
Similarly for a hyperbola the difference S'P 
S'P is constant, and equal to A A' when P is 
on one branch, and the difference SP' S'P' 
is constant and equal to A A' when P' is on 
the other branch. But no corresponding 
point seemed to exist for the parabola. 

Finally 500 years later the last great Greek 
geometer, Pappus of Alexandria, discovered 
the final secret which completed this line of 
thought. In the diagrams 16 and 18 will be 
seen two lines, XN and X'N', and in diagram 
17 the single line, XN. These are the direc- 
trices of the curves, two each for the ellipse 
and the hyperbola, and one for the parabola. 
Each directrix corresponds to its nearer focus. 
The characteristic property of a focus, S, and 
its corresponding directrix, XN, for any one 
of the three types of curve, is that the ratio 



136 INTRODUCTION TO MATHEMATICS 

(rrr>\ 
i.e. pr) is constant, where PN is 



the perpendicular on the directrix fromP, 
and P is any point on the curve. Here we 
have finally found the desired property of the 
curves which does not require us to leave 
the plane, and is stated uniformly for all 

OIT> 

three curves. For ellipses the ratio* ^^ is less 

PN 

than 1, for parabolas it is equal to 1, and for 
hyperbolas it is greater than 1. 

When Pappus had finished his investiga- 
tions, he must have felt that, apart from 
minor extensions, the subject was practically 
exhausted ; and if he could have foreseen 
the history of science for more than a thousand 
years, it would have confirmed his belief. 
Yet in truth the really fruitful ideas in con- 
nection with this branch of mathematics had 
not yet been even touched on, and no one 
had guessed their supremely important ap- 
plications in nature. No more impressive 
warning can be given to those who would 
confine knowledge and research to what is 
apparently useful, than the reflection that 
conic sections were studied for eighteen hun- 
dred years merely as an abstract science, 
without a thought of any utility other than 
to satisfy the craving for knowledge on the 
part of mathematicians, and that then at the 
end of this long period of abstract study, they 

* Cf. Note B, p. 250. 



CONIC SECTIONS 137 

were found to be the necessary key with 
which to attain the knowledge of one of the 
most important laws of nature. 

Meanwhile the entirely distinct study of 
astronomy had been going forward. The 
great Greek astronomer Ptolemy (died 168 
A.D.) published his standard treatise on the 
subject in the University of Alexandria, ex- 
plaining the apparent motions among the 
fixed stars of the sun and planets by the con- 
ception of the earth at rest and the sun and 
the planets circling round it. During the 
next thirteen hundred years the number and 
the accuracy of the astronomical observa- 
tions increased, with the result that the de- 
scription of the motions of the planets on 
Ptolemy's hypothesis had to be made more 
and more complicated. Copernicus (born 
1473 A.D. and died 1543 A.D.) pointed out 
that the motions of these heavenly bodies 
could be explained in a simpler manner if the 
sun were supposed to rest, and the earth and 
planets were conceived as moving round it. 
However, he still thought of these motions as 
essentially circular, though modified by a set 
of small corrections arbitrarily superimposed 
on the primary circular motions. So the 
matter stood when Kepler was born at Stutt- 
gart in Germany in 1571 A.D. There were 
two sciences, that of the geometry of conic 
sections and that of astronomy, both of which 



138 INTRODUCTION TO MATHEMATICS 

had been studied from a remote antiquity 
without a suspicion of any connection be- 
tween the two. Kepler was an astronomer, 
but he was also an able geometer, and on the 
subject of conic sections had arrived at ideas 
in advance of his time He is only one of 
many examples of the falsity of the idea that 
success in scientific research demands an ex- 
clusive absorption in one narrow line of study. 
Novel ideas are more apt to spring from 
an unusual assortment of knowledge not 
necessarily from vast knowledge, but from a 
thorough conception of the methods and ideas 
of distinct lines of thought. It will be re- 
membered that Charles Darwin was helped 
to arrive at his conception of the law of 
evolution by reading Malthus' famous Essay 
on Population, a work dealing with a dif- 
ferent subject at least, as it was then 
thought. 

Kepler enunciated three laws of planetary 
motion, the first two in 1609, and the third 
ten years later. They are as follows : 

(1) The orbits of the planets are ellipses, 
the sun being in the focus. 

(2) As a planet moves in its orbit, the 
radius vector from the sun to the planet 
sweeps out equal areas in equal times. 

(3) The squares of the periodic times of the 
several planets are proportional to the cubes 
of their major axes. 



CONIC SECTIONS 139 

These laws proved to be only a stage to- 
wards a more fundamental development of 
ideas. Newton (born 1642 A.D. and died 
1727 A.D.) conceived the idea of universal 
gravitation, namely, that any two pieces of \ 
matter attract each other with a force pro- 
portional to the product of their masses and , 
inversely proportional to the square of their 
distance from each other. This sweeping 
general law, coupled with the three laws of 
motion which he put into their final general 
shape, proved adequate to explain all astro- 
nomical phenomena, including Kepler's laws, 
and has formed the basis of modern physics. 
Among other things he proved that comets 
might move in very elongated ellipses, or in 
parabolas, or in hyperbolas, which are nearly 
parabolas. The comets which return such\ 
as Halley's comet must, of course, move in I 
ellipses. But the essential step in the proof of 
the law of gravitation, and even in the sug- 
gestion of its initial conception, was the veri- 
fication of Kepler's laws connecting the 
motions of the planets with the theory of 
conic sections. 

From the seventeenth century onwards the 
abstract theory of the curves has shared in 
the double renaissance of geometry due to 
the introduction of coordinate geometry and 
of projective geometry. In pEpjeetuta-gee* 
metry the fundamental ideas cluster round 



140 INTRODUCTION TO MATHEMATICS 

the consideration of sets (or pencils, as they 
are called) of lines passing through a common 
point (the vertex of the " pencil "). Now 
(cf. fig. 19) if A, B, C, Z), be any four fixed 
points on a conic section and P be a variable 
point on the curve, the pencil of lines PA, 




Fig. 19. 

PB, PC, and PD, has a special property, 
known as the constancy of its cross ratio. It 
will suffice here to say that cross ratio is a 
fundamental idea in projective geometry. 
For projective geometry this is really the de- 
finition of the curves, or some analogous pro- 
perty which is really equivalent to it, It 



CONIC SECTIONS 141 

will be seen how far in the course of ages of 
study we have drifted away from the old 
original idea of the sections of a circular cone. 
We know now that the Greeks had got hold 
of a minor property of comparatively slight 
importance ; though by some divine good 
fortune the curves themselves deserved all 
the attention which was paid to them. This 
unimportance of the '* section " idea is now 
marked in ordinary mathematical phrase- 
ology by dropping the word from their 
names. As often as not, they are now 
named merely " conies " instead of " conic 
sections." 

Finally, we come back to the point at 
which we left coordinate geometry in the last 
chapter. We had asked what was the type 
of loci corresponding to the general algebraic 
form ax-\-by=c, and had found that it was 
the class of straight lines in the plane. We 
had seen that every straight line possesses an 
equation of this form, and that every equation 
of this form corresponds to a straight line. 
We now wish to go on to the next general 
type of algebraic forms. This is evidently 
to be obtained by introducing terms involv- 
ing x 2 and xy and y 2 . Thus the new general 
form must be written 



x +2fy +c =0 
What does this represent ? The answer is 



142 INTRODUCTION TO MATHEMATICS 

that (when it represents any locus) it always re- 
presents a conic section, and, furthermore, 
that the equation of every conic section can 
always be put into this shape. The discrimi- 
nation of the particular sorts of conies as given 
by this form of equation is very easy. It en- 
tirely depends upon the consideration of ab 
h 2 , where a, b, and h, are the " constants " as 
written above. If abh 2 is a positive number, 
the curve is an ellipse ; if abh 2 ~0, the curve 
is a parabola : and if abh 2 is a negative 
number, the curve is a hyperbola. 

For example, put a =6=1, h=g=f=Q, 
c= 4. We then get the equation x 2 -\-y 2 4 
=0. It is easy to prove that this is the equa- 
tion of a circle, whose centre is at the origin, 
and radius is 2 units of length. Now abh 2 
becomes 1x1 O 2 , that is, 1, and is therefore 
positive. Hence the circle is a particular 
case of an ellipse, as it ought to be. Genera- 
lising, the equation of any circle can be 
put into the form a(x 2 -\-y 2 } -\-2gcc -\-2fy+c=0. 
Hence abh 2 becomes a 2 0, that is, a 2 , 
which is necessarily positive. Accordingly 
all circles satisfy the condition for ellipses. 
The general form of the equation of a para- 
bola is 



so that the terms of the second degree, as 



CONIC SECTIONS 143 

they are called, can be written as a perfect 
square. For squaring out, we get 

d z x 2 +2dexy +e 2 y 2 +2gx+2fy +c ; 

so that by comparison a=d 2 , h=de, b=e z , 
and therefore ab h 2 =d 2 e 2 (de) 2 =0. Hence 
the necessary condition is automatically satis- 
fied. The equation 2xy 4=0, where a =b 
=g=/=0, h=l, c= 4, represents a hyper- 
bola. For the condition abh 2 becomes 
I 2 , that is, 1, which is negative. 

The limitation, introduced by saying that, 
when the general equation represents any locus, 
it represents a conic section, is necessary, be- 
cause some particular cases of the general 
equation represent no real locus. For ex- 
ample x 2 -\-y 2 +I=Q can be satisfied by no 
real values of x and y. It is usual to say that 
the locus is now one composed of imaginary 
points. But this idea of imaginary points in 
geometry is really one of great complexity, 
which we will not now enter into. 

Some exceptional cases are included in the 
general form of the equation which may not 
be immediately recognized as conic sections. 
By properly choosing the constants the equa- 
tion can be made to represent two straight 
lines. Now two intersecting straight lines 
may fairly be said to come under the Greek 
idea of a conic section. For, by referring to 



144 INTRODUCTION TO MATHEMATICS 

the picture of the double cone above, it will 
be seen that some planes through the vertex, 
F, will cut the cone in a pair of straight lines 
intersecting at V. The case of two parallel 
straight lines can be included by considering 
a circular cylinder as a particular case of a 
cone. Then a plane, which cuts it and is 
parallel to its axis, will cut it in two parallel 
straight lines. Anyhow, whether or no the 
ancient Greek would have allowed these 
special cases to be called conic sections, they 
are certainly included among the curves re- 
presented by the general algebraic form of 
the second degree. This fact is worth noting ; 
for it is characteristic of modern mathematics 
to include among general forms all sorts of 
particular cases which would formerly have 
received special treatment. This is due to 
its pursuit of generality. 



CHAPTER XI 

FUNCTIONS 

THE mathematical use of the term function 
has been adopted also in common life. For 
example, " His temper is a function of his 
digestion," uses the term exactly in this 
mathematical sense. It means that a rule 
can be assigned which will tell you what his 
temper will be when you know how his 
digestion is working. Thus the idea of a 
" function " is simple enough, we only have 
to see how it is applied in mathematics to 
variable numbers. Let us think first of some 
concrete examples : If a train has been travel- 
ling at the rate of twenty miles per hour, the 
distance (s miles) gone after any number of 
hours, say t, is given by s=20xt; and s is 
called a function of t. Also 20 xt is thTTunc^ 
tion of t with which s is identical. If John 
is one year older than Thomas, then, when 
Thomas is at any age of x years, John's age 
(y years) is given by y=x+I ; and y is a 
function of #, namely, is the function x+\. 

In these examples t and x are called the 

145 



146 INTRODUCTION TO MATHEMATICS 

" arguments " of the functions in which they 
appear. Thus t is the argument of the func- 
tion 20 xt, and x is the argument of the func- 
tion #-{-1. If s=20xt, and t/=a?+l, tAen ^ 
and y are called the "values " of the functions 
20 xt and a?-f-l respectively. 

Coming now to the general case, we can 
define a function in mathematics as a corre- 
lation between two variable numbers, called 
respectively the argument and the value of 
the function, such that whatever value be 
assigned to the " argument of the function " 
the " value of the function " is definitely 
(i.e. uniquely) determined. The converse 
is not necessarily true, namely, that when 
the value of the function is determined 
the argument is also uniquely determined. 
Other functions of the argument x are yx 2 , 
t/=2a? 2 -|-3#+l, y=x, y=log x, y=sin x. The 
last two functions of this group will be 
readily recognizable by those who understand 
a little algebra and trigonometry. It is not 
worth while to delay now for their explana- 
tion, as they are merely quoted for the sake 
of example. 

Up to this point, though we have defined 
what we mean by a function in general, we 
have only mentioned a series of special func- 
tions. But mathematics, true to its general 
methods of procedure, symbolizes the general 
idea of any function. It does this by writing 



FUNCTIONS 147 

f\x), /(#), g(x), <f>(x), etc., for any function of 
x, where the argument x is placed in a bracket, 
and some letter like F, /, g, <f>, etc., is prefixed 
to the bracket to stand for the function. 
This notation has its defects. Thus it obvi- 
ously clashes with the convention that the 
single letters are to represent variable num- 
bers ; since here F, /, g, <, etc., prefixed to a 
bracket stand for variable functions. It 
would be easy to give examples in which we 
can only trust to common sense and the con- 
text to see what is meant. One way of 
evading the confusion is by using Greek 
letters (e.g. <f> as above) for functions ; an- 
other way is to keep to / and F (the initial 
letter of function) for the functional letter, 
and, if other variable functions have to be 
symbolized, to take an adjacent letter like g. 
With these explanations and cautions, we 
write yf(x) t to denote that y is the value of 
some undetermined function of the argument 
x ; where f(x) may stand for anything such 
as a?+l, x 2 2#4-l, sin x, log x, or merely for 
x itself. The essential point is that when x 
is given, then y is thereby definitely deter- 
mined. It is important to be quite clear as 
to the generality of this idea. Thus in y = 
f(x), we may determine, if we choose, f(x) to 
mean that when x is an integer, f(x) is zero, 
and when x has any other value, f(x) is 1. 
Accordingly, putting t/ =/(#), with this choice 



148 INTRODUCTION TO MATHEMATICS 

for the meaning of /, y is either or 1 accord- 
ing as the value of x is integral or otherwise. 
Thus /(1)=0, /(2)=0, /()=!, /(V2)=l, and 
so on. This choice for the meaning of f(x) 
gives a perfectly good function of the argu- 
ment x according to the general definition of 
a function. 

A function, which after all is only a sort 
of correlation between two variables, is re- 
presented like other correlations by a graph, 
that is in effect by the methods of coordinate 
geometry. For example, fig. 2 in Chapter II. 

is the graph of the function - where v is the 

argument and p the value of the function. 
In this case the graph is only drawn for 
positive values of v, which are the only values 
possessing any meaning for the physical ap- 
plication considered in that chapter. Again 
in fig. 14 of Chapter IX. the whole length of 
the line AB, unlimited in both directions, is 
the graph of the function aj+l, where x is the 
argument and y is the value of the function ; 
and in the same figure the unlimited line 
AiB is the graph of the function 1 x, and 
the line LOL' is the graph of the function x, 
x being the argument and y the value of the 
function. 

These functions, which are expressed by 
simple algebraic formulae, are adapted for re- 
presentation by graphs. But for some func- 



FUNCTIONS 



149 



tions this representation would be very 
misleading without a detailed explanation, or 
might even be impossible. Thus, consider the 
function mentioned above, which has the value 
1 for all values of its argument x, except 
those which are integral, e.g. except for #=0, 
o:=l, #=2, etc., when it has the value 0. 
Its appearance on a graph would be that of 
the straight line ABA' drawn parallel to the 



A' 



I 




B, 


B 2 


B 3 


B 4 


BS 



12345 



Fig. 20. 

axis XOX' at a distance from it of 1 unit of 
length. But the points, B, Ci, C%, 3, C^, etc., 
corresponding to the values 0, 1, 2, 3, 4, etc., of 
the argument x, are to be omitted, and in- 
stead of them the points 0, B\, B%, B& B, etc., 
on the axis OX, are to be taken. It is easy 
to find functions for which the graphical re- 
presentation is not only inconvenient but 
impossible. Functions which do not lend 
themselves to graphs are important in the 



150 INTRODUCTION TO MATHEMATICS 

higher mathematics, but we need not concern 
ourselves further about them here. 
* The most important division between func- 
tions is that between continuous and discon- 
tinuous functions. A function is continuous 
when its value only alters gradually for 
gradual alterations of the argument, and is 
discontinuous when it can alter its value by 
sudden jumps. Thus the two functions #+1 
and 1x, whose graphs are depicted as 
straight lines in fig. 14 of Chapter IX., are con- 
tinuous functions, and so is the function -, 

depicted in Chapter II., if we only think of 
positive values of v. But the function de- 
picted in fig. 20 of this chapter is discontinuous 
since at the values o?=l, x=2, etc., of its 
argument, its value gives sudden jumps. 

Let us think of some examples of functions 
presented to us in nature, so as to get into 
our heads the real bearing of continuity and 
discontinuity. Consider a train in its journey 
along a railway line, say from Euston Station, 
the terminus in London of the London and 
North- Western Railway. Along the line in 
order lie the stations of Bletchley and Rugby. 
Let t be the number of hours which the train 
has been on its journey from Euston, and s be 
the number of miles passed over. Then * is 
a function of t, i.e. is the variable value 
corresponding to the variable argument t. 



FUNCTIONS 151 

If we know the circumstances of the train's 
run, we know s as soon as any special value 
of t is given. Now, miracles apart, we may 
confidently assume that s is a continuous 
function of t. It is impossible to allow for 
the contingency that we can trace the train 
continuously from Euston to Bletchley, and 
that then, without any intervening time, how- 
ever short, it should appear at Rugby. The 
idea is too fantastic to enter into our calcula- 
tion : it contemplates possibilities not to be 
found outside the Arabian Nights ; and even 
in those tales sheer discontinuity of motion 
hardly enters into the imagination, they do 
not dare to tax our credulity with anything 
more than very unusual speed. But unusual 
speed is no contradiction to the great law of 
continuity of motion which appears to hold 
in nature. Thus light moves at the rate of 
about 190,000 miles per second and comes to 
us from the sun in seven or eight minutes ; 
but, in spite of this speed, its distance travelled 
is always a continuous function of the time. 
It is not quite so obvious to us that the 
velocity of a body is invariably a continuous 
function of the time. Consider the train at 
any time t : it is moving with some definite 
velocity, say v miles per hour, where v is 
zero when the train is at rest in a station and 
is negative when the train is backing. Now 
we readily allow that v cannot change its 



152 INTRODUCTION TO MATHEMATICS 

value suddenly for a big, heavy train. The 
train certainly cannot be running at forty 
miles per hour from 11.45 a.m. up to noon, 
and then suddenly, without any lapse of time, 
commence running at 50 miles per hour. We 
at once admit that the change of velocity 
will be a gradual process. But how about 
sudden blows of adequate magnitude ? Sup- 
pose two trains collide ; or, to take smaller 
objects, suppose a man kicks a football. It 
certainly appears to our sense as though the 
football began suddenly to move. Thus, in 
the case of velocity our senses do not revolt 
at the idea of its being a discontinuous func- 
tion of the time, as they did at the idea of the 
train being instantaneously transported from 
Bletchley to Rugby. As a matter of fact, 
if the laws of motion, with their conception 
of mass, are true, there is no such thing as 
discontinuous velocity in nature. Anything 
that appears to our senses as discontinuous 
change of velocity must, according to them, 
be considered to be a case of gradual change 
which is too quick to be perceptible to us. 
It would be rash, however, to rush into the 
generalization that no discontinuous functions 
are presented to us in nature. A man who, 
trusting that the mean height of the land 
above sea-level between London and Paris 
was a continuous function of the distance 
from London, walked at night on Shakes- 



FUNCTIONS 



153 



peare's Cliff by Dover in contemplation of 
the Milky Way, would be dead before he had 
had time to rearrange his ideas as to the 
necessity of caution in scientific conclusions. 
It is very easy to find a discontinuous 
function, even if we confine ourselves to the 




simplest of the algebraic formulae. For ex- 
ample, take the function y=-, which we 

x 

have already considered in the form p=-> 
where v was confined to positive values. But 



154 INTRODUCTION TO MATHEMATICS 

now let x have any value, positive or negative. 
The graph of the function is exhibited in fig. 
21. Suppose x to change continuously from 
a large negative value through a numerically 
decreasing set of negative values up to 0, and 
thence through the series of increasing posi- 
tive values. Accordingly, if a moving point, 
M, represents x on XOX', M starts at the 
extreme left of the axis XOX' and succes- 
sively moves through MI, MZ, MZ, M*, etc. 
The corresponding points on the function are 
PI, P2, PS, P-i, etc. It is easy to see that 
there is a point of discontinuity at #=0, i.e. 
at the origin O. For the value of the function 
on the negative (left) side of the origin be- 
comes endlessly great, but negative, and the 
function reappears on the positive (right) 
side as endlessly great but positive. Hen.ce, 
however small we take the length M 2 M 3 , 
there is a finite jump between the values of 
the function at M% and M 3 . Indeed, this case 
has the peculiarity that the smaller we take the 
length between M% and MS, so long as they 
enclose the origin, the bigger is the jump in 
value of the function between them. This 
graph brings out, what is also apparent in 
fig. 20 of this chapter, that for many functions 
the discontinuities only occur at isolated 
points, so that by restricting the values of the 
argument we obtain a continuous function for 
these remaining values. Thus it is evident 



FUNCTIONS 155 

from fig. 21 that in y =-, if we keep to positive 

x 

values only and exclude the origin, we obtain 
a continuous function. Similarly the same 
function, if we keep to negative values only, 
excluding the origin, is continuous. Again 
the function which is graphed in fig. 20 is con- 
tinuous between B and Ci, and between C\ 
and 2, and between C% and C& and so on, 
always in each case excluding the end points. 
It is, however, easy to find functions such thaT? 
their discontinuities occur at all points. For j 
example, consider a function /(#), such thaf^ 
when x is any fractional number /(#)=!, and 
when x is any incommensurable number 
/(#) =2. This function is discontinuous at all 
points. 

Finally, we will look a little more closely 

at the definition of continuity given above. 

We have said that a function is continuous 

when its value only alters gradually for 

gradual alterations of the argument, and is 

discontinuous when it can alter its value by 

sudden jumps. This is exactly the sort of 

definition which satisfied our mathematical 

i forefathers and no longer satisfies modern 

.mathematicians. It is worth while to spend 

ji some time over it ; for when we understand 

the modern objections to it, we shall have 

:gone a long way towards the understanding 

of the spirit of modern mathematics. The 



156 INTRODUCTION TO MATHEMATICS 

whole difference between the older and the 
newer mathematics lies in the fact that vague 
half -metaphorical terms like " gradually " 
are no longer tolerated in its exact statements. 
Modern mathematics will only admit state- 
ments and definitions and arguments which 
exclusively employ the few simple ideas about 
number and magnitude and variables on 
which the science is founded. Of two num- 
bers one can be greater or less than the 
other ; and one can be such and such a multi- 
ple of the other ; but there is no relation of 
" graduality " between two numbers, and 
hence the term is inadmissible. Now this 
may seem at first sight to be great pedantry. 
To this charge there are two answers. In 
the first place, during the first half of the 
nineteenth century it was found by some 
great mathematicians, especially Abel in 
Sweden, and Weierstrass in Germany, that 
large parts of mathematics as enunciated in 
the old happy-go-lucky manner were simply 
wrong. Macaulay in his essay on Bacon 
contrasts the certainty of mathematics with 
the uncertainty of philosophy ; and by way 
of a rhetorical example he says, " There has 
been no reaction against Taylor's theorem." 
He could not have chosen a worse example. 
For, without having made an examination of 
English text-books on mathematics contem- 
porary with the publication of this essay, the 



FUNCTIONS 157 

assumption is a fairly safe one that Taylor's 
theorem was enunciated and proved wrongly 
in every one of them. Accordingly, the 
anxious precision of modern mathematics is 
necessary for accuracy. In the second place 
it is necessary for research. It makes for 
clearness of thought, and thence for boldness 
of thought and for fertility in trying new 
combinations of ideas. When the initial 
statements are vague and slipshod, at every 
subsequent stage of thought common sense 
has to step in to limit applications and to 
explain meanings. Now in creative thought 
common sense is a bad master. Its sole 
criterion for judgment is that the new ideas 
shall look like the old ones. In other words 
it can only act by suppressing originality. 

In working our way towards the precise 
definition of continuity (as applied to func- 
tions) let us consider more closely the state- 
ment that there is no relation of " graduality " 
between numbers. It may be asked, Cannot 
one number be only slightly greater than 
another number, or in other words, cannot 
the difference between the two numbers be 
small ? The whole point is that in the ab- 
stract, apart from some arbitrarily assumed 
application, there is no such thing as a great 
or a small number. A million miles is a 
small number of miles for an astronomer 
investigating the fixed stars, but a million 



158 INTRODUCTION TO MATHEMATICS 

pounds is a large yearly income. Again, one- 
quarter is a large fraction of one's income to 
give away in charity, but is a small fraction 
of it to retain for private use. Examples can 
be accumulated indefinitely to show that 
great or small in any absolute sense have no 
abstract application to numbers. We can 
say'bf two numbers that one is greater or 
smaller than another, but not without speci- 
fication of particular circumstances that any 
one number is great or small. Our task 
therefore is to define continuity without any 
mention of a " small " or " gradual " change 
in value of the function. 

In order to do this we will give names to 
some ideas, which will also be useful when 
we come to consider limits and the differential 
calculus. 

An " interval " of values of the argument 
a? of a function /(#) is all the values lying 
between some two values of the argument. 
For example, the interval between x=l and 
#=2 consists of all the values which x can 
take lying between 1 and 2, i.e. it consists of 
all the real numbers between 1 and 2. But 
the bounding numbers of an interval need 
not be integers. An interval of values of the 
argument contains a number a, when a is a 
member of the interval. For example, the 
interval between 1 and 2 contains f , f , , and 
so on. 



FUNCTIONS 159 

A set of numbers approximates to a num- 
ber a within a standard k, when the numerical 
difference between a and every number of the 
set is less than k. Here k is the " standard 
of approximation." Thus the set of num- 
bers 3, 4, 6, 8, approximates to the number 
5 within the standard 4. In this case the 
standard 4 is not the smallest which could 
have been chosen, the set also approximates 
to 5 within any of the standards 3-1 or 3 -01 
or 3-001. Again, the numbers, 3-1, 3-141, 
3-1415, 3-14159 approximate to 3-13102 with- 
in the standard -032, and also within the 
smaller standard -03103. 

These two ideas of an interval and of 
approximation to a number within a standard 
are easy enough ; their only difficulty is that 
they look rather trivial. But when combined 
with the next idea, that of the " neighbour- 
hood " of a number, they form the foundation 
of modern mathematical reasoning. What 
do we mean by saying that something is true 
for a function f(x) in the neighbourhood of 
the value a of the argument x ? It is this 
fundamental notion which we have now got to 
make precise. 

The values of a function f(x) are said to 
possess a characteristic in the " neighbour- 
hood of a " when some interval can be found, 
which (i) contains the number a not as an 
end-point, and (ii) is such that every value 



160 INTRODUCTION TO MATHEMATICS 

of the function for arguments, other than a, 
lying within that interval possesses the char- 
acteristic. The value /(a) of the function for 
the argument a may or may not possess the 
characteristic. Nothing is decided on this 
point by statements about the neighbourhood 
of a. 

For example, suppose we take the particu- 
lar function x 2 . Now in the neighbourhood of 
2, the values of x 2 are less than 5. For we can 
find an interval, e.g. from 1 to 2 f l, which 
(i) contains 2 not as an end-point, and (ii) is 
such that, for values of x lying within it, x 2 
is less than 5. 

Now, combining the preceding ideas we 
know what is meant by saying that in the 
neighbourhood of a the function f(x) approxi- 
mates to c within the standard k. It means 
that some interval can be found which (i) 
includes a not as an end-point, and (ii) is such 
that all values of /(#), where x lies in the inter- 
val and is nota, differ fromc by less than k. For 
example, in the neighbourhood of 2, the func- 
tion i/x approximates to 1-41425 within the 
standard '0001. This is true because the 
square root of 1-99996164 is 1-4142 and the 
square root of 2*00024449 is 1-4143 ; hence 
for values of x lying in the interval 
1-99996164 to 2-00024449, which contains 2 
not as an end-point, the values of the function 
<x all lie between 1-4142 and 1-4143, and 



FUNCTIONS 161 

they therefore all differ from 1 '41425 by less 
than '0001. In this case we can, if we like, 
fix a smaller standard of approximation,^ 
namely '000051 or -0000501. Again, to take 
another example, in the neighbourhood of 2 
the function x 2 approximates to 4 within the 
standard -5. For (1'9) 2 =3'61 and (2 1)2 = 
4*41, and thus the required interval 1*9 to 
2'1, containing 2 not as an end-point, has 
been found. This example brings out the 
fact that statements about a function f(x) in 
the neighbourhood of a number a are distinct 
from statements about the value of f(x) when 
x =a. The production of an interval, through- 
out which the statement is true, is required. 
Thus the mere fact that 2 2 =4 does not by 
itself justify us in saying that in the neigh- 
bourhood of 2 the function x 2 is equal to 4. 
This statement would be untrue, because no 
interval can be produced with the required 
property. Also, the fact that 2 2 =4 does not 
by itself justify us in saying that in the 
neighbourhood of 2 the function x 2 approxi- 
mates to 4 within the standard '5 ; although 
as a matter of fact, the statement has just 
been proved to be true. 

If we understand the preceding ideas, we 
understand the foundations of modern 
mathematics. We shall recur to analogous 
ideas in the chapter on Series, and again 
in the chapter on the Differential Calculus. 



162 INTRODUCTION TO MATHEMATICS 

Meanwhile, we are now prepared to define 
" continuous functions." A function /(a?) 
is " continuous " at a value a of its argu- 
ment, when in the neighbourhood of a 
its values approximate to /(a) (i.e. to its 
value at a) within every standard of ap- 
proximation. 

This means that, whatever standard k be 
chosen, in the neighbourhood of a j(x) ap- 
proximates to /(a) within the standard k. 
For example, x 2 is continuous at the value 2 
of its argument, #, because however k be 
chosen we can always find an interval, which 
(i) contains 2 not as an end-point, and (ii) is 
such that the values of x 2 for arguments lying 
within it approximate to 4 (i.e. 2 2 ) within 
the standard k. Thus, suppose we choose 
the standard -1 ; now (1'999) 2 =3'996001, 
and (2 -01 ) 2 = 4*0401, and both these numbers 
differ from 4 by less than *1. Hence, within 
the interval 1-999 to 2 -01 the values of x 2 
approximate to 4 within the standard !. 
Similarly an interval can be produced for any 
other standard which we like to try. 

Take the example of the railway train. Its 
velocity is continuous as it passes the signal 
box, if whatever velocity you like to assign 
(say one-millionth of a mile per hour) an in- 
terval of time can be found extending before 
and after the instant of passing, such that at 
all instants within it the train's velocity 



FUNCTIONS 163 

differs from that with which the train passed 
the box by less than one-millionth of a mile 
per hour ; and the same is true whatever 
other velocity be mentioned in the place of 
one-millionth of a mile per hour. 



CHAPTER XII 

PERIODICITY IN NATURE 

THE whole life of Nature is dominated by 
the existence of periodic events, that is, by 
the existence of successive events so analogous 
to each other that, without any straining of 
language, they may be termed recurrences of 
the same event. The rotation of the earth 
produces the successive days. It is true that 
each day is different from the preceding days, 
however abstractly we define the meaning of 
a day, so as to exclude casual phenomena. 
But with a sufficiently abstract definition of 
a day, the distinction in properties between 
two days becomes faint and remote from 
practical interest ; and each day may then 
be conceived as a recurrence of the phenome- 
non of one rotation of the earth. Again the 
path of the earth round the sun leads to the 
yearly recurrence of the seasons, and imposes 
another periodicity on all the operations of 
nature. Another less fundamental perio- 
dicity is provided by the phases of the moon. 
In modern civilized life, with its artificial light, 
these phases are of slight importance, but in 

164 



PERIODICITY IN NATURE 165 

ancient times, in climates where the days are 
burning and the skies clear, human life was 
apparently largelyinfluencedby the existenceof 
moonlight. Accordingly our divisions into 
weeks and months, with their religious associa- 
tions, have spread over the European races from 
Syria and Mesopotamia, though independent 
observances following the moon's phases are 
found amongst most nations. It is, however, 
through the tides, and not through its phases 
of light and darkness, that the moon's perio- 
dicity has chiefly influenced the history of 
the earth. 

Our bodily life is essentially periodic. 
It is dominated by the beatings of the 
heart, and the recurrence of breathing. 
The presupposition of periodicity is indeed 
fundamental to our very conception of life. 
We cannot imagine a course of nature in 
which, as events progressed, we should be 
unable to say : " This has happened before." 
The whole conception of experience as a guide 
to conduct would be absent. Men would 
always find themselves in new situations 
possessing no substratum of identity with 
anything in past history. The very means of 
measuring time as a quantity would be absent. 
Events might still be recognized as occurring 
in a series, so that some were earlier and 
others later. But we now go beyond this 
bare recognition. We can not only say that 



16 INTRODUCTION TO MATHEMATICS 

three events, A, B, C, occurred in this order, 
so that A came before B, and B before C ; 
but also we can say that the length of time 
between the occurrences of A and B was 
twice as long as that between B and C. Now, 
quantity of time is essentially dependent on 
observing the number of natural recurrences 
which have intervened. We may say 
that the length of time between A and B was 
so many days, or so many months, or so 
many years, according to the type of recur- 
rence to which we wish to appeal. Indeed, 
at the beginning of civilization, these three 
modes of measuring time were really distinct. 
It has been one of the first tasks of science 
among civilized or semi-civilized nations, to 
fuse them into one coherent measure. The 
full extent of this task must be grasped. It 
is necessary to determine, not merely what 
number of days (e.g. 365 '25 . . .) go to some 
one year, but also previously to determine that 
the same number of days do go to the suc- 
cessive years. We can imagine a world in 
which periodicities exist, but such that no two 
are coherent. In some years there might be 
200 days and in others 350. The determina- 
tion of the broad general consistency of the 
more important periodicities was the first step 
in natural science. This consistency arises 
from no abstract intuitive law of thought ; 
it is merely an observed fact of nature 



PERIODICITY IN NATURE 167 

guaranteed by experience. Indeed, so far is 
it from being a necessary law, that it is not 
even exactly true There are divergencies in 
every case. For some instances these diver- 
gencies are easily observed and are therefore 
immediately apparent. In other cases it re- 
quires the most refined observations and 
astronomical accuracy to make them appar- 
ent. Broadly speaking, all recurrences de- 
pending on living beings, such as the beatings 
of the heart, are subject in comparison with 
other recurrences to rapid variations. The 
great stable obvious recurrences stable in 
the sense of mutually agreeing with great 
accuracy are those depending on the motion 
of the earth as a whole, and on similar motions 
of the heavenly bodies. 

We therefore assume that these astronomi- 
cal recurrences mark out equal intervals of 
time. But how are we to deal with their 
discrepancies which the refined observations 
of astronomy detect ? Apparently we are 
reduced to the arbitrary assumption that one 
or other of these sets of phenomena marks out 
equal times e.g. that either all days are of 
equal length, or that all years are of equal 
length. This is not so : some assumptions 
must be made, but the assumption which 
underlies the whole procedure of the astrono- 
mers in determining the measure of time is 
that the laws of motion are exactly verified. 



Before explaining how this is done, it is in- 
teresting to observe that this relegation of 
the determination of the measure of time to 
the astronomers arises (as has been said) from 
the stable consistency of the recurrences with 
which they deal. If such a superior con- 
sistency had been noted among the recur- 
rences characteristic of the human body, we 
should naturally have looked to the doctors 
of medicine for the regulation of our clocks. 

In considering how the laws of motion 
come into the matter, note that two incon- 
sistent modes of measuring time will yield 
different variations of velocity to the same 
body. For example, suppose we define an 
hour as one twenty-fourth of a day, and take 
the case of a train running uniformly for two 
hours at the rate of twenty miles per hour. 
Now take a grossly inconsistent measure of 
time, and suppose that it makes the first hour 
to be twice as long as the second hour. Then, 
according to this other measure of duration, 
the time of the train's run is divided into 
two parts, during each of which it has tra- 
versed the same distance, namely, twenty 
miles ; but the duration of the first part is 
twice as long as that of the second part. 
Hence the velocity of the train has not been 
uniform, and on the average the velocity 
during the second period is twice that during 
the first period. Thus the question as to 



PERIODICITY IN NATURE 169 

whether the train has been running uniformly 
or not entirely depends on the standard of 
time which we adopt. 

Now, for all ordinary purposes of life on the 
earth, the various astronomical recurrences 
may be looked on as absolutely consistent ; 
and, furthermore assuming their consistency, 
and thereby assuming the velocities and 
changes of velocities possessed by bodies, we 
find that the laws of motion, which have 
been considered above, are almost exactly 
verified. But only almost exactly when we 
come to some of the astronomical phenomena. 
We find, however, that by assuming slightly 
different velocities for the rotations and 
motions of the planets and stars, the laws 
would be exactly verified. This assumption 
is then made ; and we have, in fact thereby, 
adopted a measure of time, which is indeed 
defined by reference to the astronomical 
phenomena, but not so as to be consistent 
with the uniformity of any one of them. But 
the broad fact remains that the uniform flow 
of time on which so much is based, is itself 
dependent on the observation of periodic 
events. 

Even phenomena, which on the surface 
seem casual and exceptional, or, on the other 
hand, maintain themselves with a uniform 
persistency, may be due to the remote influ- 
ence of periodicity. Take for example, the 



170 INTRODUCTION TO MATHEMATICS 

principle of resonance. Resonance arises 
when two sets of connected circumstances 
have the same periodicities. It is a dynami- 
cal law that the small vibrations of all bodies 
when left to themselves take place in definite 
times characteristic of the body. Thus a 
pendulum with a small swing always vibrates 
in some definite time, characteristic of its shape 
and distribution of weight and length. A more 
complicated body may have many ways of 
vibrating ; but each of its modes of vibration 
will have its own peculiar " period." Those 
periods of vibration of a body are called its 
** free " periods. Thus a pendulum has but 
one period of vibration, while a suspension 
bridge will have many. We get a musical 
instrument, like a violin string, when the 
periods of vibration are all simple submultiples 
of the longest ; i.e. if t seconds be the longest 
period, the others are \t, \t, and so on, where 
any of these smaller periods may be absent. 
Now, suppose we excite the vibrations of a 
body by a cause which is itself periodic; 
then, if the period of the cause is very nearly 
that of one of the periods of the body, that 
mode of vibration of the body is very violently 
excited ; even although the magnitude of the 
exciting cause is small. This phenomenon is 
called " resonance." The general reason is 
easy to understand. Any one wanting to 
upset a rocking stone will push " in tune " 



PERIODICITY IN NATURE 171 

with the oscillations of the stone, so as always 
to secure a favourable moment for a push. 
If the pushes are out of tune, some increase 
the oscillations, but others check them. But 
when they are in tune, after a time all the 
pushes are favourable. The word " reson- 
ance " comes from considerations of sound : 
but the phenomenon extends far beyond the 
region of sound. The laws of absorption and 
emission of light depend on it, the " tuning " 
of receivers for wireless telegraphy, the com- 
parative importance of the influences of 
planets on each other's motion, the danger 
to a suspension bridge as troops march over 
it in step, and the excessive vibration of some 
ships under the rhythmical beat of their 
machinery at certain speeds. This coinci- 
dence of periodicities may produce steady 
phenomena when there is a constant associ- 
ation of the two periodic events, or it may 
produce violent and sudden outbursts when 
the association is fortuitous and temporary. 
Again, the characteristic and constant 
periods of vibration mentioned above are 
the underlying causes of what appear to 
us as steady excitements of our senses. We 
work for hours in a steady light, or we listen 
to a steady unvarying sound. But, if modern 
science be correct, this steadiness has no 
counterpart in nature. The steady light is 
due to the impact on the eye of a countless 



172 INTRODUCTION TO MATHEMATICS 

number of periodic waves in a vibrating ether, 
and the steady sound to similar waves in a 
vibrating air. It is not our purpose here to 
explain the theory of light or the theory of 
sound. We have said enough to make it 
evident that one of the first steps necessary 
to make mathematics a fit instrument for the 
investigation of Nature is that it should be 
able to express the essential periodicity of 
things. If we have grasped this, we can 
understand the importance of the mathe- 
matical conceptions which we have next to 
consider, namely, periodic functions. 



CHAPTER XIII 

TRIGONOMETRY 

TRIGONOMETRY did not take its rise from 
the general consideration of the periodicity of 
nature. In this respect its history is analo- 
gous to that of conic sections, which also had 
their origin in very particular ideas. Indeed, 
a comparison of the histories of the two 
sciences yields some very instructive analogies 
and contrasts. Trigonometry, like conic sec- 
tions, had its origin among the Greeks. Its 
inventor was Hipparchus (born about 160 
B.C.), a Greek astronomer, who made his 
observations at Rhodes. His services to 
astronomy were very great, and it left his 
hands a truly scientific subject with important 
results established, and the right method of 
progress indicated. Perhaps the invention 
of trigonometry was not the least of these 
services to the main science of his study. The 
next man who extended trigonometry was 
Ptolemy, the great Alexandrian astronomer, 
whom we have already mentioned. We now 

173 



174 INTRODUCTION TO MATHEMATICS 

see at once the great contrast between conic 
sections and trigonometry. The origin of 
trigonometry was practical ; it was invented 
because it was necessary for astronomical re- 
search. The origin of conic sections was 
purely theoretical. The only reason for its 
initial study was the abstract interest of the 
ideas involved. Characteristically enough 
conic sections were invented about 150 years 
earlier than trigonometry, during the very 
best period of Greek thought. But the im- 
portance of trigonometry, both to the theory 
and the application of mathematics, is only 
one of innumerable instances of the fruitful 
ideas which the general science has gained 
from its practical applications. 

We will try and make clear to ourselves 
what trigonometry is, and why it should be 
generated by the scientific study of astronomy. 
In the first place : What are the measure- 
ments which can be made by an astronomer ? 
They are measurements of time and measure- 
ments of angles. The astronomer may adjust 
a telescope (for it is easier to discuss the 
familiar instrument of modern astronomers) 
so that it can only turn about a fixed axis 
pointing east and west ; the result is that 
the telescope can only point to the south, with 
a greater or less elevation of direction, or, if 
turned round beyond the zenith, point to the 
north. This is the transit instrument, the 



TRIGONOMETRY 175 

great instrument for the exact measurement 
of the times at which stars are due south or 
due north. But indirectly this instrument 
measures angles. For when the time elapsed 
between the transits of two stars has been 
noted, by the assumption of the uniform 
rotation of the earth, we obtain the angle 
through which the earth has turned in that 
period of time. Again, by other instruments, 
the angle between two stars can be directly 
measured. For if E is the eye of the astrono- 




Fig. 22. 

mer, and EA and EB are the directions in 
which the stars are seen, it is easy to devise 
instruments which shall measure the angle 
AEB. Hence, when the astronomer is form- 
ing a survey of the heavens, he is, in fact, 
measuring angles so as to fix the relative 
directions of the stars and planets at any in- 
stant. Again, in the analogous problem of 



176 INTRODUCTION TO MATHEMATICS 

land-surveying, angles are the chief subject 
of measurements. The direct measurements 
of length are only rarely possible with any 
accuracy ; rivers, houses, forests, mountains, 
and general irregularities of ground all get in 
the way. The survey of a whole country will 
depend only on one or two direct measure- 
ments of length, made with the greatest 
elaboration in selected places like Salisbury 
Plain. The main work of a survey is the 
measurement of angles. For example, A, B, 
and C will be conspicuous points in the dis- 




Fig. 23. 

trict surveyed, say the tops of church towers. 
These points are visible each from the others. 
Then it is a very simple matter at A to 
measure the angle BAG, and at B to measure 
the angle ABC, and at C to measure the angle 
BCA. Theoretically, it is only necessary to 
measure two of these angles ; for, by a well- 
known proposition in geometry, the sum of 
the three angles of a triangle amounts to two 



TRIGONOMETRY 177 

right-angles, so that when two of the angles 
are known, the third can be deduced. It is 
better, however, in practice to measure all 
three, and then any small errors of observa- 
tion can be checked. In the process of map- 
making a country is completely covered with 
triangles in this way. This process is called 
triangulation, and is the fundamental process 
in a survey. 

Now, when all the angles of a triangle are 
known, the shape of the triangle is known 
that is, the shape as distinguished from the 
size. We here come upon the great principle 
of geometrical similarity. The idea is very 
familiar to us in its practical applications. 
We are all familiar with the idea of a plan 
drawn to scale. Thus if the scale of a plan 
be an inch to a yard, a length of three inches 
in the plan means a length of three yards in 
the original. Also the shapes depicted in the 
plan are the shapes in the original, so that a 
right-angle in the original appears as a right- 
angle in the plan. Similarly in a map, which 
is only a plan of a country, the proportions 
of the lengths in the map are the proportions 
of the distances between the places indicated, 
and the directions in the map are the direc- 
tions in the country. For example, if in the 
map one place is north-north-west of the 
other, so it is in reality ; that is to say, in a 
map the angles are the same as in reality. 



178 INTRODUCTION TO MATHEMATICS 

Geometrical similarity may be defined thus : 
Two figures are similar (i) if to any point 
in one figure a point in the other figure 
corresponds, so that to every line there is a 
corresponding line, and to every angle a 
corresponding angle, and (ii) if the lengths 
of corresponding lines are in a fixed propor- 
tion, and the magnitudes of corresponding 
angles are the same. The fixed proportion 
of the lengths of corresponding lines in a map 
(or plan) and in the original is called the scale 
of the map. The scale should always be 
indicated on the margin of every map and 
plan. It has already been pointed out that 
two triangles whose angles are respectively 
equal are similar. Thus, if the two triangles 





6 E' C F 

Fig. 24. 

ABC and DEF have the angles at A and D 
equal, and those at B and E, and those at C 
and F, then DE is to AB in the same propor- 



TRIGONOMETRY 



179 



tion as EF is to BC, and as FD is to CA. 

But it is not true of other figures that simi- 
larity is guaranteed by the mere equality of 
angles. Take for example, the familiar cases 
of a rectangle and a square. Let ABCD be 
a square, and ABEF be a rectangle. Then 
all the corresponding angles are equal. But 



B 



Fig. 25. 



whereas the side AB of the square is equal to 
the side AB of the rectangle, the side EC of 
the square is about half the size of the side 
BE of the rectangle. Hence it is not true 
that the square ABCD is similar to the rect- 
angle ABEF. This peculiar property of the 
triangle, which is not shared by other recti- 
linear figures, makes it the fundamental 
figure in the theory of similarity. Hence in 
surveys, triangulation is the fundamental 
process ; and hence also arises the word " tri- 



180 INTRODUCTION TO MATHEMATICS 

gonometry," derived from the two Greek 
words trigonon a triangle and metria measure- 
ment. The fundamental question from which 
trigonometry arose is this : Given the magni- 
tudes of the angles of a triangle, what can be 
stated as to the relative magnitudes of the 
sides. Note that we say " relative magnitudes 
of the sides," since by the theory of similarity 
it is only the proportions of the sides which 
are known. In order to answer this ques- 
tion, certain functions of the magnitudes of 
an angle, considered as the argument, are in- 
troduced. In their origin these functions 
were got at by considering a right-angled tri- 
angle, and the magnitude of the angle was 
defined by the length of the arc of a circle. 
In modern elementary books, the funda- 
mental position of the arc of the circle as de- 
fining the magnitude of the angle has been 
pushed somewhat to the background, not to 
the advantage either of theory or clearness 
of explanation. It must first be noticed 
that, in relation to similarity, the circle holds 
the same fundamental position among curvi- 
linear figures, as does the triangle among 
rectilinear figures. Any two circles are simi- 
lar figures ; they only differ in scale. The 
lengths of the circumferences of two circles, 
such as APA' and A-J*\A\ in the fig. 26 are 
in proportion to the lengths of their radii. 
Furthermore, if the two circles have the same 



TRIGONOMETRY 



181 



centre 0, as do the two circles in fig. 26, then 
the arcs AP and A\P\ intercepted by the 
arms of any angle AOP, are also in propor- 
tion to their radii. Hence the ratio of the 




Fig. 26. 

length of the arc AP to the length of the 

arc A.P 

radius OP, that is 7^= is a number which 
radius OP 

is quite independent of the length OP, and is 

the same as the fraction -^ ^4- . This f rac- 

radms OPi 

tion of " arc divided by radius " is the proper 
theoretical way to measure the magnitude of 



182 INTRODUCTION TO MATHEMATICS 

an angle ; for it is dependent on no arbitrary 
unit of length, and on no arbitrary way of 
dividing up any arbitrarily assumed angle, 

AP 

such as a right-angle. Thus the fraction 



represents the magnitude of the angle AOP. 
Now draw PM perpendicularly to OA. Then 
the Greek mathematicians called the line PM 
the sine of the arc AP, and the line OM the 
cosine of the arc AP. They were well aware 
that the importance of the relations of these 
various lines to each other was dependent on 
the theory of similarity which we have just 
expounded. But they did not make their 
definitions express the properties which arise 
from this theory. Also they had not in their 
heads the modern general ideas respecting 
functions as correlating pairs of variable num- 
bers, nor in fact were they aware of any 
modern conception of algebra and algebraic 
analysis. Accordingly, it was natural to 
them to think merely of the relations between 
certain lines in a diagram. For us the case 
is different : we wish to embody our more 
powerful ideas. 

Hence, in modern mathematics, instead 
of considering the arc AP, we consider 

AP 

the fraction , which is a number the 



same for all lengths of OP ; and, instead of 
considering the lines PM and OM, we con- 



TRIGONOMETRY 183 

PM , OM , . , 
sider the fractions and -, which again 



are numbers not dependent on the length of 
OP, i.e. not dependent on the scale of our 

PM 

diagrams. Then we define the number 



PA 

to be the sine of the number -, and the 



number =- to be the cosine of the number 



PA 

These fractional forms are clumsy to 



AP 

print ; so let us put u for the fraction 



which represents the magnitude of the angle 

PM 
AOP, and put v for the fraction -, and w 



OM 

for the fraction - Then u, v, w, are num- 



bers, and, since we are talking of any angle 
AOP, they are variable numbers. But al 
correlation exists between their magnitudes,! 
so that when u (i.e. the angle AOP) is given, 
the magnitudes of v and w are definitely deter-! 
mined. Hence v and w are functions of the 
argument u. We have called v the sine of 
u, and w the cosine of u. We wish to adapt 
the general functional notation y=f(x) to 
these special cases : so in modern mathe- 
matics we write sin for " / " when we want to 



184 INTRODUCTION TO MATHEMATICS 

indicate the special function of " sine," and 
"cos" for "/" when we want to indicate 
the special function of "cosine." Thus, with 
the above meanings for u, v, w, we get 

v=sin u, and w=cos u, 

where the brackets surrounding the x in /(#) 
are omitted for the special functions. The 
meaning of these functions sin and cos as 
correlating the pairs of numbers u and v, and 
u and w is, that the functional relations are to 
be found by constructing (cf. fig. 26) an angle 
AOP, whose measure " AP divided by OP " 
is equal to u, and that then v is the number 
given by " PM divided by OP " and w is the 
number given by " OM divided by OP." 

It is evident that without some further defi- 
nitions we shall get into difficulties when the 
number u is taken too large. For then the arc 
AP may be greater than one-quarter of the 
circumference of the circle, and the point M 
(cf. figs. 26 and 27) may fall between and A' 
and not between O and A. Also P may be 
below the line AOA' and not above it as in 
fig. 26. In order to get over this difficulty 
we have recourse to the ideas and conven- 
tions of coordinate geometry in making our 
complete definitions of the sine and cosine. 
Let one arm OA of the angle be the axis 
OX, and produce the axis backwards to 
obtain its negative part OX'. Draw the 



TRIGONOMETRY 



185 



other axis YOY' perpendicular to it. Let 
any point P at a distance r from O have 
coordinates x and y. These coordinates are 
both positive in the first "quadrant" of 
the plan, e.g. the coordinates x and y of 




P in fig. 27. In the other quadrants, either 
one or both of the coordinates are negative, 
for example, x' and y for P', and x' and y' 
for P", and x and y' for P'" in fig. 27, where 
a?' and y' are both negative numbers. The 
positive angle POA is the arc AP divided 

by r, its sine is - and its cosine is - ; the posi- 



tive angle AOP' is the arc ABP' divided by r, 
its sine is - and cosine - ; the positive angle 

AOP" is the arc ABA'P" divided by r, its 

t/' x' 

sine is - and its cosine is - ; the positive 
r r 

angle AOP'" is the arc ABA'B'P'" divided 

w' ^E 

by r, its sine is - and its cosine is -. 
r r 

But even now we have not gone far enough. 
For suppose we choose t* to be a number 
greater than the ratio of the whole circum- 
ference of the circle to its radius. Owing to 
the similarity of all circles this ratio is the 
same for all circles. It is always denoted in 
mathematics by the symbol 2ir t where TT 
is the Greek form of the letter p and its 
name in the Greek alphabet is " pi." It can 
be proved that TT is an incommensurable 
number, and that therefore its value cannot 
be expressed by any fraction, or by any 
terminating or recurring decimal. Its value 
to a few decimal places is 3-14159 ; for many 
purposes a sufficiently accurate approximate 

value is . Mathematicians can easily cal- 

7 

culate TT to any degree of accuracy required, 
just as \/2 can be so calculated. Its value 
has been actually given to 707 places of 



TRIGONOMETRY 187 

decimals. Such elaboration of calculation is 
merely a curiosity, and of no practical or 
theoretical interest. The accurate deter^ 
mination of TT is one of the two parts of 
the famous problem of squaring the circle. 
The other part of the problem is, by the 
theoretical methods of pure geometry to 
describe a straight line equal in length to the 
circumference. Both parts of the problem 
are now known to be impossible ; and the 
insoluble problem has now lost all special 
practical or theoretical interest, having be- ' 
come absorbed in wider ideas. 

After this digression on the value of TT, we 
now return to the question of the general 
definition of the magnitude of an angle, so as 
to be able to produce an angle corresponding 
to any value u. Suppose a moving point, Q, 
to start from A on OX (cf . fig. 27), and to rotate 
in the positive direction (anti-clockwise, in 
the figure considered) round the circumference 
of the circle for any number of times, finally 
resting at any point, e.g. at P or P' or P" or 
P"'. Then the total length of the curvilinear 
circular path traversed, divided by the radius 
of the circle, r, is the generalized definition of ( 
a positive angle of any size. Let x, y be the 
coordinates of the point in which the point Q 
rests, i.e.in one of the four alternative positions 
mentioned in fig. 27 ; x and y (as here used) will 
either x and y, or x r and y, or x' and t/', or x 



188 INTRODUCTION TO MATHEMATICS 
and y'. Then the sign of this generalized 

a i flfl 

angle is - and its cosine is -. With these 
r r 

definitions the functional relations a=sin u 
and ry=cos u, are at last defined for all posi- 
tive real values of u. For negative values of 
u we simply take rotation of Q in the opposite 
(clockwise) direction ; but it is not worth our 
while to elaborate further on this point, now 
that the general method of procedure has 
been explained. 

These functions of sine and cosine, as thus 
defined, enable us to deal with the problems 
concerning the triangle from which Trigono- 
metry took its rise. But we are now in a 
position to relate Trigonometry to the wider 
idea of Periodicity of which the importance 
was explained in the last chapter. It is easy 
to see that the functions sin u and cos u are 
periodic functions of u. For consider the 
position, P (in fig. 27), of a moving point, Q, 
which has started from A and revolved round 
the circle. This position, P, marks the angles 

arc AP , , . arc AP , A . arc AP 
- , and 2 TT-\ --- , and 4 ir-\ -- , 



Q Trt 

and 6 TT + , and so on indefinitely. Now, 
all these angles have the same sine and cosine, 

?/ 72 

namely, and -. Hence it is easy to see that, 



TRIGONOMETRY 189 

if u be chosen to have any value, the argu- 
ments u and 2 TT+U, and 4-77+ w, and 67r-fw, 
and STT+U and so on indefinitely, have all the 
same values for the corresponding sines and 
cosines. In other words, 



sn w=sn 2 
=etc. ; 
COS U = COS (27T+tt)=COS (47T+w)=COS 

=etc. 

This fact is expressed by saying that s in u and 
cos u are periodic functions with their period 
equal to 2?r. 

The graph of the function t/=sin x (notice 
that we now abandon v and u for the more 
familiar y and x) is shown in fig. 28. We take 
on the axis of x any arbitrary length at pleasure 
to represent the number TT, and on the axis 
of y any arbitrary length at pleasure to -repre- 
sent the number 1. The numerical values of 
the sine and cosine can never exceed unity. 
The recurrence of the figure after periods of 
2-7T will be noticed. This graph represents the 
simplest style of periodic function, out of 
which all others are constructed. The cosine 
gives nothing fundamentally different from the 
sine. For it is easy to prove that cos x= 

rjf 

sin (# + -) ; hence it can be seen that the 

it 

graph of cos x is simply fig. 28 modified by 



190 INTRODUCTION TO MATHEMATICS 

drawing the axis of OF through the point 
on OX marked -, instead of drawing it in 

its actual position on the figure. 

It is easy to construct a ' sine ' function in 




Fig. 28. 

which the period has any assigned value a. 
For we have only to write 

. 27TX 



and then 



sin 



(27TX . _ 
l -- \-2ir 
[a 



I . STHC 

1= sin . 

d 



Thus the period of this new function is now a. 
Let us now give a general definition of what 



TRIGONOMETRY 191 

we mean by a periodic function. The function 
/(a?) is periodic, with the period a, if (i) for any 
value of x we have f(x)=f(x-{-a), and (ii) there 
is no number b smaller than a such that for 
any value of x, f(x)=f(x-}-b). 

The second clause is put into the definition 

because when we have sin - , it is not only 

a 

periodic in the period a, but also in the periods 
2a and 3a, and so on ; this arises since 



. 27r(#+3a) . /2-rrx , \ . 2irx 
sin - ! - ' = sml -- h67r 1 =sm - . 
a \ a / a 

So it is the smallest period which we want to 
get hold of and call the period of the function. 
The greater part of the abstract theory of 
periodic functions and the whole of the appli- 
cations of the theory to Physical Science are 
dominated by an important theorem called 
Fourier's Theorem ; namely that, if /(#) be a 
periodic function with the period a and if f(x) 
also satisfies certain conditions, which practic- 
ally are always presupposed in functions sug- 
gested by natural phenomena, then /(a?) can 
be written as the sum of a set of terms in the 
form 

2TTX . \ , . /47T 

-f j j +c 2 sin \r 



. iG-rrx , \ 
in I -- 1-^3) 



. , , 

+c 3 sin I -- 1-^3)+ etc. 



192 INTRODUCTION TO MATHEMATICS 

In this formula CQ, GI, c%, 03, etc., and also 
fit %> 3* etc., are constants, chosen so as to 
suit the particular function. Again we have 
to ask, How many terms have to be chosen ? 
And here a new difficulty arises : for we can 
prove that, though in some particular cases a 
definite number will do, yet in general all we 
can do is to approximate as closely as we like 
to the value of the function by taking more 
and more terms. This process of gradual 
approximation brings us to the consideration 
of the theory of infinite series, an essential 
part of mathematical theory which we will 
consider in the next chapter. 

The above method of expressing a periodic 
function as a sum of sines is called the " har- 
monic analysis " of the function. For ex- 
ample, at any point on the sea coast the tides 
rise and fall periodically. Thus at a point 
near the Straits of Dover there will be two 
daily tides due to the rotation of the earth. 
The daily rise and fall of the tides are com- 
plicated by the fact that there are two tidal 
waves, one coming up the English Channel, 
and the other which has swept round the 
North of Scotland, and has then come south- 
ward down the North Sea. Again some high 
tides are higher than others : this is due to 
the fact that the Sun has also a tide-generating 
influence as well as the Moon. In this way 
monthly and other periods are introduced. 



TRIGONOMETRY 193 

We leave out of account the exceptional in- 
fluence of winds which cannot be foreseen. 
The general problem of the harmonic analysis 
of the tides is to find sets of terms like those 
in the expression on page 191 above, such that 
each set will give with approximate accuracy 
the contribution of the tide-generating influ- 
ences of one " period " to the height of the 
tide at any instant. The argument x will 
therefore be the time reckoned from any con- 
venient commencement. 

Again, the motion of vibration of a violin 
string is submitted to a similar harmonic 
analysis, and so are the vibrations of the 
ether and the air, corresponding respectively 
to waves of light and waves of sound. We 
are here in the presence of one of the funda- 
mental processes of mathematical physics 
namely, nothing less than its general method 
of dealing with the great natural fact of 
Periodicity. 



CHAPTER XIV 

SERIES 

No part of Mathematics suffers more from 
the triviality of its initial presentation to 
beginners than the great subject of series. 
Two minor examples of series, namely arith- 
metic and geometric series, are considered ; 
these examples are important because they 
are the simplest examples of an important 
general theory. But the general ideas are 
never disclosed ; and thus the examples, which 
exemplify nothing, are reduced to silly triviali- 
ties. 

The general mathematical idea of a series 
is that of a set of things ranged in order, that 
is, in sequence; This meaning is accurately 
represented in the common use of the term. 
Consider for example, the series of English 
Prime Ministers during the nineteenth century, 
arranged in the order of their first tenure of 
that office within the century. The series 
commences with William Pitt, and ends with 
Lord Rosebery, who, appropriately enough, 
is the biographer of the first member. We 

194 



SERIES 195 

might have considered other serial orders for 
the arrangement of these men ; for example, 
according to their height or their weight. 
These other suggested orders strike us as 
trivial in connection with Prime Ministers, 
and would not naturally occur to the mind ; 
but abstractly they are just as good orders 
as any other. When one order among terms 
is very much more important or more obvious 
than other orders, it is often spoken of as the 
order of those terms. Thus the order of 'the 
integers would always be taken to mean their 
order as arranged in order of magnitude. But 
of course there is an indefinite number -of 
other ways of arranging them. When the 
number of things considered is finite, the 
number of ways of arranging them in order is 
called the number of their permutations. The 
number of permutations of a set of n things, 
where n is some finite integer, is 

nx(n l)x(n 2)x(n 3)x...x4x3x2xl 

that is to say, it is the product of the first n 
integers ; this product is so important in 
mathematics that a special symbolism, is used 
for it, and it is always written ' n 1 ' Thus, 
21=2x1=2, and 3!=3x2xl=6, and 4!=4 
x3x2xl=24, and 51=5x4x3x2x1=120. 
As n increases, the value of n \ increases very 
quickly ; thus 100 ! is a hundred times as 
large as 99 I 



196 INTRODUCTION TO MATHEMATICS 

It is easy to verify in the case of small 
values of n that n ! is the number of ways 
of arranging n things in order. Thus con- 
sider two things a and b ; these are capable 
of the two orders ab and ba, and 2 ! =2. 

Again, take three things a, b, and c ; these 
are capable of the six orders, abc, acb, bac, 
bca, cab, cba t and 31=6. Similarly for the 
twenty-four orders in which four things a, b, 
c, and d, can be arranged. 

When we come to the infinite sets of things 
like the sets of all the integers, or all the 
fractions, or all the real numbers for instance 
we come at once upon the complications of 
the theory of order-types. This subject was 
touched upon in Chapter VI. in considering 
the possible orders of the integers, and of the 
fractions, and of the real numbers. The 
whole question of order-types forms a com- 
paratively new branch of mathematics of 
great importance. We shall not consider it 
any further. All the infinite series which we 
consider now are of the same order-type as 
the integers arranged in ascending order of 
magnitude, namely, with a first term, and 
such that each term has a couple of next- 
door neighbours, one on either side, with the 
exception of the first term which has, of 
course, only one next-door neighbour. Thus, 
if m be any integer (not zero), there will be 
always an mth term. A series with a finite 



SERIES 197 

number of terms (say n terms) has the same 
characteristics as far as next-door neighbours 
are concerned as an infinite series ; it only 
differs from infinite series in having a last 
term, namely, the nth. 

The important thing to do with a series of 
numbers using for the future " series " in 
the restricted sense which has just been men- 
tioned is to add its successive terms to- 
gether. 

Thus if u\, Uz, 3, . . . u n . . . are respec- 
tively the 1st, 2nd, 3rd, 4th, . . . nth, . . . 
terms of a series of numbers, we form succes- 
sively the series u\ t u\+uz, ^1+^2+^3, i-f- 
W2+W3+W4, and so on ; thus the sum of the 
1st n terms may be written. 



If the series has only a finite number of 
terms, we come at last in this way to the 
sum of the whole series of terms. But, if 
the series has an infinite number of terms, 
this process of successively forming the sums 
of the terms never terminates ; and in this 
sense there is no such thing as the sum of an 
infinite series. 

But why is it important successively to add 
the terms of a series in this way ? The answer 
is that we are here symbolizing the funda- 
mental mental process of approximation. 
This is a process which has significance far 



198 INTRODUCTION TO MATHEMATICS 

beyond the regions of mathematics. Our 
limited intellects cannot deal with compli- 
cated material all at once, and our method of 
arrangement is that of approximation. The 
statesman in framing his speech puts the 
dominating issues first and lets the details 
fall naturally into their subordinate places. 
There is, of course, the converse artistic 
method of preparing the imagination by the 
presentation of subordinate or special details, 
and then gradually rising to a crisis. In 
either way the process is one of gradual sum- 
mation of effects ; and this is exactly what 
is done by the successive summation of the 
terms of a series. Our ordinary method of 
stating numbers is such a process of gradual 
summation, at least, in the case of large 
numbers. Thus 568,213 presents itself to 
the mind as 

500,000 +60,000 +8,000 +200 +10 +3 

In the case of decimal fractions this is so 
more avowedly. Thus 3-14159 is 



Also, 3 and 3+^, and 3+^+^, and 

~t~Ttfff ~^T?nn7 anc * ^+T 1 o'i'T^4~T7nn7~l~TTmn7 are 
successive approximations to the complete re- 

sult 3-14159. If we read 568,218 backwards 
from right to left, starting with the 3 units, 



SERIES 199 

we read it in the artistic way, gradually pre- 
paring the mind for the crisis of 500,000. 

The ordinary process of numerical multi- 
plication proceeds by means of the summa- 
tion of a series, Consider the computation 



342 

658 



2736 
1710 
2052 

225036 

Hence the three lines to be added form a 
series of which the first term is the upper 
line. This series follows the artistic method 
of presenting the most important term last, 
not from any feeling for art, but because of 
the convenience gained by keeping a firm 
hold on the units' place, thus enabling us to 
omit some O's, formally necessary. 

But when we approximate by gradually 
adding the successive terms of an infinite 
series, what are we approximating to ? The 
difficulty is that the series has no " sum " in 
the straightforward sense of the word, because 
the operation of adding together its terms 
can never be completed. The answer is that 
we are approximating to the limit of the 
summation of the series, and we must now 



200 INTRODUCTION TO MATHEMATICS 

proceed to explain what the " limit " of a 
series is. 

The summation of a series approximates to 
a limit when the sum of any number of its 
terms, provided the number be large enough, 
is as nearly equal to the limit as you care to 
approach. But this description of the mean- 
ing of approximating to a limit evidently will 
not stand the vigorous scrutiny of modern 
mathematics. What is meant by large 
enough, and by nearly equal, and by care to 
approach ? All these vague phrases must be 
explained in terms of the simple abstract 
ideas which alone are admitted into pure 
mathematics. 

Let the successive terms of the series be 
i, U2, Ws, W4, . . . , u n) etc., so that u n is the 
nth term of the series. Also let s n be the 
sum of the 1st n terms, whatever n may be. 
So that 



and 



Then the terms $1, $2, $3, . . . $ n , . . . form 
a new series, and the formation of this series 
is the process of summation of the original 
series. Then the " approximation " of the 
summation of the original series to a " limit " 
means the " approximation of the terms of 
this new series to a limit." And we have 



SERIES 201 

now to explain what we mean by the approxi- 
mation to a limit of the terms of a series. 

Now, remembering the definition (given in 
chapter XII.) of a standard of approxima- 
tion, the idea of a limit means this : I is 
the limit of the terms of the series si, $2, 
*s s n , . . ., if, corresponding to each 
real number k, taken as a standard of 
approximation, a term s n of the series can 
be found so that all succeeding terms (i.e. 
s n+i> *n+2> e * c -) approximate to I within 
that standard of approximation. If another 
smaller standard k 1 be chosen, the term 
s n may be too early in the series, and a 
later term 8 m with the above property will 
then be found. 

If this property holds, it is evident that as 
you go along to series Si, $2, $3, . . ., s n , . . . 
from left to right, after a time you come to 
terms all of which are nearer to I than any 
number which you may like to assign. In 
other words you approximate to I as closely 
as you like. The close connection of this 
definition of the limit of a series with the 
definition of a continuous function given in 
chapter XI. will be immediately perceived. 

Then coming back to the original series MI, 
t*2 3, . . . u n , . . ., the limit of the terms of 
the series Si, $2, $3, . . , 9 s n , . . ., is called 
the " sum to infinity " of the original series. 
But it is evident that this use of the word 



202 INTRODUCTION TO MATHEMATICS 

" sum " is very artificial, and we must not 
assume the analogous properties to those of 
the ordinary sum of a finite number of terms 
without some special investigation. 

Let us look at an example of a " sum to 
infinity." Consider the recurring decimal 
1111. . . . This decimal is merely a way of 
symbolizing the "sum to infinity " of the series 
1, -01, -001, -0001, etc. The correspond- 
ing series found by summation is si = -I t 
$2 ='11, 53 ='111, 54 =-1111, etc. The limit 
of the terms of this series is ; this is easy to 
see by simple division, for 

^=a+ 7 V=-ll+^=.lll+ irzr Vir= etc. 
Hence, if T 3 T is given (the k of the definition), 
1 and all succeeding terms differ from by 
less than T 8 T ; if -^^ is given (another choice 
for the k of the definition), -111 and all 
succeeding terms differ from by less than 
YflVo-; and so on, whatever choice for k be 
made. 

It is evident that nothing that has been 
said gives the slightest idea as to how the 
"sum to infinity" of a series is to be 
found. We have merely stated the condi- 
tions which such a number is to satisfy. In- 
deed, a general method for finding in all 
cases the sum to infinity of a series is intrinsic- 
ally out of the question, for the simple reason 
that such a " sum," as here defined, does not 
always exist. Series which possess a sum to 



SERIES 203 

infinity are called convergent, and those which 
do not possess a sum to infinity are called 
divergent. 

An obvious example of a divergent series 
is 1, 2, 3, . . ., n . . . i.e. the series of in- 
tegers in their order of magnitude. For 
whatever number I you try to take as its 
sum to infinity, and whatever standard ol 
approximation A; you choose, by taking 
enough terms of the series you can always 
make their sum differ from / by more than 
k. Again, another example of a divergent 
series is 1, 1, 1, etc., i.e. the series ol 
which each term is equal to 1. Then the 
sum of n terms is n, and this sum grows 
without limit as n increases. Again, another 
example of a divergent series is 1, 1, 1, 1, 
1, 1, etc., i.e. the series in which the terms 
are alternately 1 and 1. The sum of an 
odd number of terms is 1, and of an even 
number of terms is 0. Hence the terms of 
the series $1, $2, $3, . . . s n , . . . do not ap- 
proximate to a limit, although they do not 
increase without limit. 

It is tempting to suppose that the condi- 
tion for MI, 2 u nt . . . to have a sum 
to infinity is that u n should decrease inde- 
finitely as n increases. Mathematics would 
be a much easier science than it is, if this 
were the case. Unfortunately the supposition 
is not true. 



204 INTRODUCTION TO MATHEMATICS 

For example the series 

111 1 

7 2' 3' 4' ' ' *' n ' * ' 

is divergent. It is easy to see that this is 
the case ; for consider the sum of n terms 
ginning at the (n+1)" 1 term. These n 



4-U 

terms are - -, - -, - -, ...--: there 
w+l'n+2'n+3' 2n 

are n of them and is the least among them. 

SBO 

Hence their sum is greater than n times 
, i.e. is greater than -. Now, without 

altering the sum to infinity, if it exist, we 
can add together neighbouring terms, and 
obtain the series 



that is, by what has been said above, a series 
whose terms after the 2nd are greater than 
those of the series, 

1, i, i, I, etc., 

where all the terms after the first are equal. 
But this series is divergent. Hence the 
original series is divergent.* 

This question of divergency shows how 
careful we must be in arguing from the pro- 

* Cf. Note C, p. 251. 



SERIES 205 

perties of the sum of a finite number of terms 
to that of the sum of an infinite series. For 
the most elementary property of a finite 
number of terms is that of course they 
possess a sum : but even this fundamental 
property is not necessarily possessed by an 
infinite series. This caution merely states 
that we must not be misled by the suggestion 
of the technical term " sum of an infinite 
series." It is usual to indicate the sum of 
the infinite series 

t*i, u 2 , t*s, . . . t* n . . . . by 



We now pass on to a generalization of the 
idea of a series, which mathematics, true to 
its method, makes by use of the variable. 
Hitherto, we have only contemplated series 
in which each definite term was a definite 
number. But equally well we can generalize, 
and make each term to be some mathematical 
expression containing a variable x. Thus 
we may consider the series 1, x, x 2 , x* t . . ., 
x n t . . ., and the series 

x 2 - x 3 x" 

*' ' 8" ..... 7P ' ' ' 

In order to symbolize the general idea of 
any such function, conceive of a function of 
x, f n (x) say, which involves in its formation 
a variable integer n, then, by giving n the 



206 INTRODUCTION TO MATHEMATICS 

values 1, 2, 3, etc., in succession, we get the 
series 

/i(), h(x), h(x), . . ., f n (x), . . . 

Such a series may be convergent for some 
values of as and divergent for others. It is, 
in fact, rather rare to find a series involving a 
variable x which is convergent for all values 
of x, at least in any particular instance it is 
very unsafe to assume that this is the case. 
For example, let us examine the simplest of 
all instances, namely, the " geometrical " 
series 

1, x, x 2 , # 3 , . . ., x n , . . . 

The sum of n terms is given by 



Now multiply both sides by x and we get 



Now subtract the last line from the upper 
line and we get 

s n (l x) =s n xs n =1 aP+\ 
and hence (if x be not equal to 1) 

1 gn+l . 1 x n+l 



^ __ 

n I-x ~ l^x ~T^x 
Now if x be numerically less than 1, for suffi- 

jn+l 

ciently large values of n, - - is always numeri- 

L X J 



SERIES 20T 

cally less than k, however k be chosen. Thus, 
if x be numerically less than 1, the series 1, #, 

# 2 , . . . tc n , . . . is convergent, and - - is its 

1 x 

limit. This statement is symbolized by 

. . ., (-1 <x 



A X 

But if x' is numerically greater than 1, or 
numerically equal to 1, the series is divergent. 
In other words, if x lie between 1 and -f-l> 
the series is convergent ; but if x be equal 
to 1 or -}-l, or if a; lie outside the interval 
1 to +1, then the series is divergent. Thus 
the series is convergent at all " points " 
within the interval 1 to +i> exclusive of 
the end points. 

At this stage of our enquiry another ques- 
tion arises. Suppose that the series 



is convergent for all values of x lying within 
the interval a to b, i.e. the series is convergent 
for any value of x which is greater than a and 
less than b. Also, suppose we want to be 
sure that in approximating to the limit we 
add together enough terms to come within 
some standard of approximation k. Can we 
always state some number of terms, say n, 
such that, if we take n or more terms to 
form the sum, then whatever value x has 



208 INTRODUCTION TO MATHEMATICS 

within the interval we have satisfied the 
desired standard of approximation? 

Sometimes we can and sometimes we can- 
not do this for each value of k. When we 
can, the series is called uniformly convergent 
throughout the interval, and when we cannot 
do so, the series is called non-uniformly con- 
vergent throughout the interval. It makes 
a great difference to the properties of a series 
whether it is or is not uniformly convergent 
through an interval. Let us illustrate the 
matter by the simplest example and the 
simplest numbers. 

Consider the geometric series 



It is convergent throughout the interval 
1 to +1, excluding the end values aj= 1. 

But it is not uniformly convergent through- 
out this interval. For if s n (x) be the sum of 
n terms, we have proved that the difference 

1 # n+1 

between s n (x) and the limit - - is - - 

1 a? \x 

Now suppose n be any given number of terms, 
say 20, and let k be any assigned standard 
of approximation, say -001. Then, by taking 
$ near enough to + 1 or near enough to 1, 

#21 

we can make the numerical value of - - to 

Ia? 

be greater than -001. Thus 20 terms will 



SERIES 209 

not do iver the whole interval, though it is 
more thai enough over some parts of it. 

The sane reasoning can be applied what- 
ever other number we take instead of 20, 
and whatever standard of approximation in- 
stead of -001. Hence the geometric series 
I-\-x+x 2 -{-x 3 -t . . . +x tt + ... is non-uni- 
formly convergent over its whole interval of 
convergence 1 to -f-1. But if we take any 
smaller interval lying at both ends within the 
interval 1 to +1, the geometric series is 
uniformly convergent within it. For ex- 
ample, take the interval to +^. Then any 



value for n which makes - - numerically 

1 a; 

less than k at these limits for x also serves 
for all values of x between these limits, since 



it so happens that diminishes in numeri- 

1 x 

cal value as x diminishes in numerical value. 
For example, take k =-001; then, putting 
x = vzr . we find : 



forn=l, 



1 X 1 Tr 

-rn+1 / 1 \3 

for n=2, - = ^r =ir^= -00111 . . ., 

1 X 1 rfr 



f or n= 3, 5 = J^ = W V7= '000111 . . ., 

1 x 1 yir 

Thus three terms will do for the whole in- 



210 INTRODUCTION TO MATHEMATICS 

terval, though, of course, for some parts of 
the interval it is more than is necessary. 
Notice that, because l+a4-# 2 -}- . . . 
+ n + ... is convergent (though not uni- 
formly) throughout the interva 1 1 to +1, 
for each value of x in the internal some num- 
ber of terms n can be found wMch will satisfy 
a desired standard of approximation ; but, 
as we take x nearer and nearer to either end 
value +1 or 1, larger and larger values of 
n have to be employed. 

It is curious that this important distinction 
between uniform and non-uniform conver- 
gence was not published till 1847 by Stokes 
afterwards, Sir George Stokes and later, in- 
dependently in 1850 by Seidel, a German 
mathematician. 

The critical points, where non-uniform con- 
vergence comes in, are not necessarily at the 
limits of the interval throughout which con- 
vergence holds. This is a speciality belonging 
to the geometric series. 

In the case of the geometric series l+# 
. . +x n + . . ., a simple algebraic 



expression - - can be given for its limit in 
1 x 

its interval of convergence. But this is not 
always the case. Often we can prove a series 
to be convergent within a certain interval, 
though we know nothing more about its 
limit except that it is the limit of the series. 



SERIES 211 

But this is a very good way of defining a 
function ; viz. as the limit of an infinite con- 
vergent series, and is, in fact, the way in which 
most functions are, or ought to be, defined. 

Thus, the most important series in ele- 
mentary analysis is 



where n \ has the meaning defined earlier in 
this chapter. This series can be proved to 
be absolutely convergent for all values of , 
and to be uniformly convergent within any 
interval which we like to take. Hence it has 
all the comfortable mathematical properties 
which a series should have. It is called the 
exponential series. Denote its sum to infinity 
by exp#. Thus, by definition, 



expa? is called the exponential function. 

It is fairly easy to prove, with a little 
knowledge of elementary mathematics, that 

(expa?)x(expt/)=exp(#+t/) . . .(A) 
In other words that 



212 INTRODUCTION TO MATHEMATICS 

This property (A) is an example of what 
is called an addition-theorem. IVhen any 
function [say /(#)] has been denned, the first 
thing we do is to try to express /(#+*/) in terms 
of known functions of x only, and known func- 
tions of y only. If we can do so, the result 
is called an addition-theorem. Addition- 
theorems play a great part in mathematical 
analysis. Thus the addition-theorem for the 
sine is given by 

sin (x+y)sin x cos y+cos x sin y, 
and for the cosine by 

cos (x+y) = cos x cos y sin x sin y. 

As a matter of fact the best ways of de- 
fining sin x and cos x are not by the elaborate 
geometrical methods of the previous chapter, 
but as the limits respectively of the series 

x 3 x 5 x 



so that we put 



x 3 . x 5 x 7 . 
sin *=*--+___ +etc 

, X 2 , X 4 X Q , 

cos *=!-_+- +etc 



SERIES 



218 



These definitions are equivalent to the geo- 
metrical definitions, and both series can be 
proved to be convergent for all values of a?, 
and uniformly convergent throughout any 
interval. These series for sine and cosine 
have a general likeness to the exponential 
series given above. They are, indeed, intim- 
ately connected with it by means of the 
theory of imaginary numbers explained in 
Chapters VII. and VIII. 



X. A * 

Fig. 29. 

The graph of the exponential function is 
given in fig. 29. It cuts the axis OF at the 
point t/=l, as evidently it ought to do, since 
when x=Q every term of the series except 
the first is zero. The importance of the ex- 
ponential function is that it represents any 
changing physical quantity whose rate of 
increase at any instant is a uniform per- 
centage of its value at that instant. For 



214 INTRODUCTION TO MATHEMATICS 

example, the above graph represents the size 
at any time of a population with a uniform 
birth-rate, a uniform death-rate, and no emi- 
gration, where the x corresponds to the time 
reckoned from any convenient day, and the 
y represents the population to the proper 
scale. The scale must be such that OA re- 
presents the population at the date which is 
taken as the origin. But we have here come 
upon the idea of " rates of increase " which 
is the topic for the next chapter. 

An important function nearly allied to the 
exponential function is found by putting aj 2 
for x as the argument in the exponential func- 
tion. We thus get exp. ( x 2 ). The graph 
t/=exp. ( x 2 ) is given in fig. 30. 




Fig. 30. 



The curve, which is something like a cocked 
hat, is called the curve of normal error. Its 



SERIES 



215 



corresponding function is vitally important 
to the theory of statistics, and tells us in 
many cases the sort of deviations from the 
average results which we are to expect. 

Another important function is found by 
combining the exponential function with the 
sine, in this way : 



y =exp( car) xsin 



2772! 




Fig. 31. 

Its graph is given in fig. 31. The points 
A, B, O, C, D, E, F, are placed at equal in- 
tervals \p, and an unending series of them 
should be drawn forwards and backwards. 
This function represents the dying away of 
vibrations under the influence of friction or of 
" damping " forces. Apart from the friction, 
the vibrations would be periodic, with a 
period p ; but the influence of the friction 



216 INTRODUCTION TO MATHEMATICS 

makes the extent of each vibration smaller 
than that of the preceding by a constant per- 
centage of that extent. This combination 
of the idea of " periodicity " (which requires 
the sine or cosine for its symbolism) and of 
" constant percentage " (which requires the 
exponential function for its symbolism) is the 
reason for the form of this function, namely, 
its form as a product of a sine-function into 
an exponential function. 



CHAPTER XV 

THE DIFFERENTIAL CALCULUS 

THE invention of the differential calculus 
marks a crisis in the history of mathematics, 
The progress of science is divided between 
periods characterized by a slow accumulation 
of ideas and periods, when, owing to the new 
material for thought thus patiently collected, 
some genius by the invention of a new method 
or a new point of view, suddenly transforms 
the whole subject on to a higher level. These 
contrasted periods in the progress of the 
history of thought are compared by Shelley 
to the formation of an avalanche. 

The sun-awakened avalanche ! whose mass, 

Thrioe sifted by the storm, had gathered there 

Flake after flake, in heaven-defying minds 

As thought by thought is piled, till some great truth 

Is loosened, and the nations echo round, 

The comparison will bear some pressing. 
The final burst of sunshine which awakens 
the avalanche is not necessarily beyond com- 
parison in magnitude with the other powers 
of nature which have presided over its slow 

217 



218 INTRODUCTION TO MATHEMATICS 

formation. The same is true in science. The 
genius who has the good fortune to produce 
the final idea which transforms a whole 
region of thought, does not necessarily excel 
all his predecessors who have worked at the 
preliminary formation of ideas. In consider- 
ing the history of science, it is both silly and 
ungrateful to confine our admiration with a 
gaping wonder to those men who have made 
the final advances towards a new epoch 

In the particular instance before us, the 
subject had a long history before it as- 
sumed its final form at the hands of its 
two inventors. There are some traces of its 
methods even among the Greek mathe- 
maticians, and finally, just before the actual 
production of the subject, Fermat (born 1601 
A.D., and died 1665 A.D.), a distinguished 
French mathematician, had so improved on 
previous ideas that the subject was all but 
created by him. Fermat, also, may lay 
claim to be the joint inventor of coordinate 
geometry in company with his contemporary 
and countryman, Descartes. It was, in fact, 
Descartes from whom the world of science 
received the new ideas, but Fermat had cer- 
tainly arrived at them independently. 

We need not, however, stint our admira- 
tion either for Newton or for Leibniz. New- 
ton was a mathematician and a student of 
physical science, Leibniz was a mathema- 



DIFFERENTIAL CALCULUS 219 

tician and a philosopher, and each of them 
in his own department of thought was one of 
the greatest men of genius that the world 
has known. The joint invention was the 
occasion of an unfortunate and not very 
creditable dispute. Newton was using the 
methods of Fluxions, as he called the subject, 
in 1666, and employed it in the composition 
of his Principia, although in the work as 
printed any special algebraic notation is 
avoided. But he did not print a direct state- 
ment of his method till 1693. Leibniz pub- 
lished his first statement in 1684. He was 
accused by Newton's friends of having got 
it from a MS. by Newton, which he had been 
shown privately. Leibniz also accused New- 
ton of having plagiarized from him. There 
is now not very much doubt but that both 
should have the credit of being independent 
discoverers. The subject had arrived at a 
stage in which it was ripe for discovery, and 
there is nothing surprising in the fact that 
two such able men should have independ- 
ently hit upon it. 

These joint discoveries are quite common 
in science. Discoveries are not in general 
made before they have been led up to 
by the previous trend of thought, and by 
that time many minds are in hot pursuit 
of the important idea. If we merely keep 
to discoveries in which Englishmen are 



220 INTRODUCTION TO MATHEMATICS 

concerned, the simultaneous enunciation of 
the law of natural selection by Darwin and 
Wallace, and the simultaneous discovery of 
Neptune by Adams and the French astrono- 
mer, Leverrier, at once occur to the mind. 
The disputes, as to whom the credit ought to 
be given, are often influenced by an unworthy 
spirit of nationalism. The really inspiring 
reflection suggested by the history of mathe- 
matics is the unity of thought and interest 
among men of so many epochs, so many nations, 
and so many races. Indians, Egyptians, 
Assyrians, Greeks, Arabs, Italians, French- 
men, Germans, Englishmen, and Russians, have 
all made essential contributions to the pro- 
gress of the science. Assuredly the jealous 
exaltation of the contribution of one particu- 
lar nation is not to show the larger spirit. 

The importance of the differential calculus 
arises from the very nature of the subject, 
which is the systematic consideration of the 
rates of increase of functions. This idea is 
immediately presented to us by the study of 
nature ; velocity is the rate of increase of the 
distance travelled, and acceleration is the 
rate of increase of velocity. Thus the funda- 
mental idea of change, which is at the basis of 
our whole perception of phenomena, immedi- 
ately suggests the enquiry as to the rate of 
change. The familiar terms of " quickly " 
and " slowly " gain their meaning from a tacit 



DIFFERENTIAL CALCULUS 221 



reference to rates of change. Thus the differ- 
ential calculus is concerned with the very 
key of the position from which mathematics 
can be successfully applied to the explanation 
of the course of nature. 

This idea of the rate of change was certainly 
in Newton's mind, and was embodied in the 




o 



T ft 

Fig. 32. 



language in which he explained the subject. 
It may be doubted, however, whether this 
point of view, derived from natural phenomena, 
was ever much in the minds of the preced- 
ing mathematicians who prepared the subject 
for its birth. They were concerned with the 
more abstract problems of drawing tangents 
to curves, of finding the lengths of curves, and 
of finding the areas enclosed by curves. The 



222 INTRODUCTION TO MATHEMATICS 

last two problems, of the rectification of curves 
and the quadrature of curves as they are 
named, belong to the Integral Calculus, which 
is however involved in the same general subject 
as the Differential Calculus. 

The introduction of coordinate geometry 
makes the two points of view coalesce. For 
(cf. fig. 32) let AQP be any curved line and let 
PT be the tangent at the point P on it. Let 
the axes of coordinates be OX and OY ; and 
let y =/(#) be the equation to the curve, so that 
OM=x, and PM=y. Now let Q be any 
moving point on the curve, with coordinates 
#i */i> ; then yi =f(xi). And let Q' be the point 
on the tangent with the same abscissa x\ ; 
suppose that the coordinates of Q' are x\ and 
y'. Now suppose that N moves along the 
axis OX from left to right with a uniform 
velocity ; then it is easy to see that the ordi- 
nate y' of the point Q' on the tangent TP also 
increases uniformly as Q' moves along the 
tangent in a corresponding way. In fact it is 
easy to see that the ratio of the rate of increase 
of Q'N to the rate of increase of ON is in the 
ratio of Q'N to TN, which is the same at all 
points of the straight line. But the rate of 
increase of Q2V, which is the rate of increase 
of /(#i), varies from point to point of the curve 
so long as it is not straight. As Q passes 
through the point P, the rate of increase of 
/ (#1) (where x\ coincides with x for the moment) 



is the same as the rate of increase of y' on the 
tangent at P. Hence, if we have a general 
method of determining the rate of increase 
of a function /(#) of a variable x, we can 
determine the slope of the tangent at any 
point (x, y,) on a curve, and thence can 
draw it. Thus the problems of drawing tan- 
gents to a curve, and of determining the 
rates of increase of a function are really 
identical. 

It will be noticed that, as in the cases of 
Conic Sections and Trigonometry, the more 
artificial of the two points of view is the one 
in which the subject took its rise. The really 
fundamental aspect of the science only rose 
into prominence comparatively late in the 
day. It is a well-founded historical genera- 
lization, that the last thing to be discovered 
in any science is what the science is really 
about. Men go on groping for centuries, 
guided merely by a dim instinct and a puzzled 
curiosity, till at last " some great truth is 
loosened." 

Let us take some special cases in order to 
familiarize ourselves with the sort of ideas 
which we want to make precise. A train is 
in motion how shall we determine its velocity 
at some instant, let us say, at noon ? We can 
take an interval of five minutes which includes 
noon, and measure how far the train has gone 
in that period. Suppose we find it to be five 



224 INTRODUCTION TO MATHEMATICS 

miles, we may then conclude that the train 
was running at the rate of 60 miles per hour. 
But five miles is a long distance, and we 
cannot be sure that just at noon the train 
was moving at this pace. At noon it may 
have been running 70 miles per hour, and 
afterwards the break may have been put on. 
It will be safer to work with a smaller interval, 
say one minute, which includes noon, and to 
measure the space traversed during that 
period. But for some purposes greater 
accuracy may be required, and one minute 
may be too long. In practice, the necessary 
inaccuracy of our measurements makes it 
useless to take too small a period for measure- 
ment. But in theory the smaller the period 
the better, and we are tempted to say that 
for ideal accuracy an infinitely small period 
is required. The older mathematicians, in 
particular Leibniz, were not only tempted, 
but yielded to the temptation, and did say 
it. Even now it is a useful fashion of speech, 
provided that we know how to interpret it 
into the language of common sense. It is 
curious that, in his exposition of the founda- 
tions of the calculus, Newton, the natural 
scientist, is much more philosophical than 
Leibniz, the philosopher, and on the other 
hand, Leibniz provided the admirable nota- 
tion which has been so essential for the pro- 
gress of the subject. 



Now take another example within the region 
of pure mathematics. Let us proceed to find 
the rate of increase of the function x 2 for 
any value x of its argument. We have not 
yet really denned what we mean by rate of 
increase. We will try and grasp its meaning 
in relation to this particular case. When x 
increases to x +h, the function x 2 increases to 
(x-\-h) 2 ; so that the total increase has been 
(x-\-h) 2 x 2 , due to an increase h in the argu- 
ment. Hence throughout the interval x to 
(x+ h) the average increase of the function per 

, . (x+h) 2 -x 2 
unit increase of the argument is - ^~ - . 

But 

(x+h) 2 =x 2 +2hx+h* t 

and therefore 



nm ,, 



Thus 2x+h is the average increase of the 
function x 2 per unit increase in the argument, 
the average being taken over by the interval 
x to x+h. But 2x+h depends on h, the size 
of the interval. We shall evidently get what 
we want, namely the rate of increase at the 
value x of the argument, by diminishing h 
more and more. Hence in the limit when h 



226 INTRODUCTION TO MATHEMATICS 

has decreased indefinitely, we say that 2x is the 
rate of increase of x 2 at the value x of the 
argument. 

Here again we are apparently driven up 
against the idea of infinitely small quantities 
in the use of the words " in the limit when h 
has decreased indefinitely." Leibniz held that, 
mysterious as it may sound, there were actu- 
ally existing such things as infinitely small 
quantities, and of course infinitely small num- 
bers corresponding to them. Newton's lan- 
guage and ideas were more on the modern 
lines ; but he did not succeed in explaining 
the matter with such explicitness so as to be 
evidently doing more than explain Leibniz's 
ideas in rather indirect language. The real 
explanation of the subject was first given by 
Weierstrass and the Berlin School of mathe- 
maticians about the middle of the nineteenth 
century. But between Leibniz and Weier- 
strass a copious literature, both mathematical 
and philosophical, had grown up round these 
mysterious infinitely small quantities which 
mathematics had discovered and philosophy 
proceeded to explain. Some philosophers, 
Bishop Berkeley, for instance, correctly denied 
the validity of the whole idea, though for 
reasons other than those indicated here. But 
the curious fact remained that, despite all 
criticisms of the foundations of the subject, 
there could be no doubt but that the mathe- 



DIFFERENTIAL CALCULUS 227 

matical procedure was substantially right. In 
fact, the subject was right, though the explana- 
tions were wrong. It is this possibility of 
being right, albeit with entirely wrong ex- 
planations as to what is being done, that so 
often makes external criticism that is so far 
as it is meant to stop the pursuit of a method 
singularly barren and futile in the progress of 
science. The instinct of trained observers, 
and their sense of curiosity, due to the fact 
that they are obviously getting at something, 
are far safer guides. Anyhow the general 
effect of the success of the Differential Calculus 
was to generate a large amount of bad philo- 
sophy, centring round the idea of the in- 
finitely small. The relics of this verbiage 
may still be found in the explanations of 
many elementary mathematical text-books on 
the Differential Calculus. It is a safe rule to 
apply that, when a mathematical or philoso- 
phical author writes with a misty profundity, 
he is talking nonsense. 

Newton would have phrased the question 
by saying that, as h approaches zero, in the 
limit 2x+h becomes 2#. It is our task so to 
explain this statement as to show that it does 
not in reality covertly assume the existence 
of Leibniz's infinitely small quantities. In 
reading over the Newtonian method of state- 
ment, it is tempting to seek simplicity by 



228 INTRODUCTION TO MATHEMATICS 

saying that 2x+h is 2x, when h is zero. But 
this will not do ; for it thereby abolishes the 
interval from as to x+ h, over which the average 
increase was calculated. The problem is, how 
to keep an interval of length h over which to 
calculate the average increase, and at the same 
time to treat h as if it were zero. Newton did 
this by the conception of a limit, and we now 
proceed to give Weiers trass's explanation of 
its real meaning. 

In the first place notice that, in discussing 
2x +h, we have been considering x as fixed in 
value and h as varying. In other words x 
has been treated as a " constant " variable, 
or parameter, as explained in Chapter IX. ; 
and we have really been considering 2x+h as 
a function of the argument h. Hence we can 
generalize the question on hand, and ask 
what we mean by saying that the function 
/(/&) tends to the limit I, say, as its argument 
h tends to the value zero. But again we shall 
see that the special value zero for the argument 
does not belong to the essence of the subject ; 
and again we generalize still further, and ask, 
what we mean by saying that the function f(h) 
tends to the limit I as h tends to the value a. 

Now, according to the Weierstrassian ex- 
planation the whole idea of h tending to the 
value , though it gives a sort of metaphorical 
picture of what we are driving at, is really off 
the point entirely. Indeed it is fairly obvious 



that, as long as we retain anything like "A 
tending to a," as a fundamental idea, we are 
really in the clutches of the infinitely small ; 
for we imply the notion of h being infinitely 
near to a. This is just what we want to get 
rid of. 

Accordingly, we shall yet again restate our 
phrase to be explained, and ask what we 
mean by saying that the limit of the function 
f(h) at a is I. 

The limit of /(/) at a is a property of the 
neighbourhood of a, where " neighbourhood " 
is used in the sense defined in Chapter XI. 
during the discussion of the continuity of 
functions. The value of the function f(h) at 
a is /(a) ; but the limit is distinct in idea 
from the value, and may be different from 
it, and may exist when the value has not 
been defined. We shall also use the term 
" standard of approximation " in the sense 
in which it is defined in Chapter XI. In 
fact, in the definition of " continuity " given 
towards the end of that chapter we have 
practically defined a limit. The definition of 
a limit is : 

A function /(#) has the limit I at a value 
a of its argument a?, when in the neighbour- 
hood of a its values approximate to I within 
every standard of approximation. 

Compare this definition with that already 
given for continuity, namely : 



230 INTRODUCTION TO MATHEMATICS 

A function f(x) is continuous at a value a 
of its argument, when in the neighbourhood 
of a its values approximate to its value at a 
within every standard of approximation. 

It is at once evident that a function is con- 
tinuous at a when (i) it possesses a limit at a, 
and (ii) that limit is equal to its value at a. 
Thus the illustrations of continuity which 
have been given at the end of Chapter XI. are 
illustrations of the idea of a limit, namely, 
they were all directed to proving that /(a) 
was the limit of /(#) at a for the functions 
considered and the value of a considered. It 
is really more instructive to consider the 
limit at a point where a function is not con- 
tinuous. For example, consider the function 
of which the graph is given in fig. 20 of Chap- 
ter XI. This function /(#) is denned to have 
the value 1 for all values of the argument 
except the integers 0, 1, 2, 3, etc., and for these 
integral values it has the value 0. Now let 
us think of its limit when x=3. We notice 
that in the definition of the limit the value 
of the function at a (in this case, a =3) is ex- 
cluded. But, excluding /(3), the values of 
/(#), when on lies within any interval which 
(i) contains 3 not as an end-point, and (ii) 
does not extend so far as 2 and 4, are all 
equal to 1 ; and hence these values approxi- 
mate to 1 within every standard of approxi- 
mation. Hence 1 is the limit of /(#) at the 



DIFFERENTIAL CALCULUS 231 

value 3 of the argument x, but by definition 
/(3)=0. 

This is an instance of a function which 
possesses both a value and a limit at the 
value 3 of the argument, but the value is not 
equal to the limit. At the end of Chapter 
XI. the function x 2 was considered at the 
value 2 of the argument. Its value at 2 is 2 2 , 
i.e. 4, and it was proved that its limit is also 
4. Thus here we have a function with a 
value and a limit which are equal. 

Finally we come to the case which is essen- 
tially important for our purposes, namely, to 
a function which possesses a limit, but no 
defined value at a certain value of its argu- 
ment. We need not go far to look for 

2di 

such a function, -- will serve our purpose. 
x 

Now in any mathematical book, we might 

2x 
find the equation, - =2, written without 

x 

hesitation or comment. But there is a diffi- 
culty in this ; for when x is zero, = - ; and 

x 

- has no defined meaning. Thus the value 

2# 

of the function - - at xQ has no defined 
x 



232 INTRODUCTION TO MATHEMATICS 

meaning. But for every other value of x, 

O/V| 

the value of the function is 2. Thus the 

x 

2/p 
limit of at xQ is 2, and it has no value 

x 

x 2 
at x=0. Similarly the limit of at xa is 

a whatever a may be, so that the limit of 

a? 2 a? 2 

at 0=0 is 0. But the value of at x=0 

x x 

takes the form -, which has no denned 

#2 
meaning. Thus the function - - has a limit 

x 

but no value at 0. 

We now come back to the problem from 
which we started this discussion on the nature 
of a limit. How are we going to define the 
rate of increase of the function x 2 at any 
value x of its argument. Our answer is that 
this rate of increase is the limit of the func- 

(x -\-Tl\Z _ /r;2 

tion v ^ ' at the value zero for its 



argument h. (Note that x is here a " con- 
stant.") Let us see how this answer works 



DIFFERENTIAL CALCULUS 233 

in the light of our definition of a limit. We 
have 

(a?+A) 2 -a 2 _2/kc+A 2 _h(2x+h) 
h h h 

Now in finding the limit of ^ , ' at the 

value of the argument h, the value (if any) 
of the function at h=Q is excluded. But for 
all values of h, except A=0, we can divide 



through by h. Thus the limit of at 

h=0 is the same as that of 2x+h at h=0. 
Now, whatever standard of approximation k 
we choose to take, by considering the interval 
from \k to +\k we see that, for values of 
h which fall within it, 2x+h differs from 2x 
by less than \k, that is by less than k. This 
is true for any standard k. Hence in the neigh- 
bourhood of the value for /, 2x-\-h approxi- 
mates to 2x within every standard of approxi- 
mation, and therefore 2x is the limit of 2x+h 
at h=0. Hence by what has been said above 

2x is the limit of ; - at the value 

n 

for h. It follows, therefore, that 2x is what 
we have called the rate of increase of x 2 at 
the value x of the argument. Thus this 
method conducts us to the same rate of in- 



284 INTRODUCTION TO MATHEMATICS 

crease for x 2 as did the Leibnizian way of 
making h grow " infinitely small." 

The more abstract terms " differential co- 
efficient," or " derived function," are gener- 
ally used for what we have hitherto called the 
" rate of increase " of a function. The 
general definition is as follows : the differ- 
ential coefficient of the function f(x) is the 

limit, if it exist, of the function 



of the argument h at the value of its argu- 
ment. 

How have we, by this definition and the 
subsidiary definition of a limit, really managed 
to avoid the notion of " infinitely small num- 
bers " which so worried our mathematical 
forefathers ? For them the difficulty arose 
because on the one hand they had to use an 
interval x to x+h over which to calculate 
the average increase, and, on the other hand, 
they finally wanted to put h=0. The result 
was they seemed to be landed into the notion 
of an existent interval of zero size. Now 
how do we avoid this difficulty ? In this 
way we use the notion that corresponding 
to any standard of approximation, some in- 
terval with such and such properties can be 
found. The difference is that we have 
grasped the importance of the notion of " the 
variable," and they had not done so. Thus, 



DIFFERENTIAL CALCULUS 235 

at the end of our exposition of the essential 
notions of mathematical analysis, we are led 
back to the ideas with which in Chapter II. 
we commenced our enquiry that in mathe- 
matics the fundamentally important ideas 
are those of " some things " and ** any 
things.'? 



CHAPTER XVI 

GEOMETRY 

GEOMETRY, like the rest of mathematics, is 
abstract. In it the properties of the shapes 
and relative positions of things are studied. 
But we do not need to consider who is observ- 
ing the things, or whether he becomes ac- 
quainted with them by sight or touch or 
hearing. In short, we ignore all particular 
sensations. Furthermore, particular things 
such as the Houses of Parliament, or the 
terrestrial globe are ignored. Every pro- 
position refers to any things with such and 
such geometrical properties. Of course it 
helps our imagination to look at particular 
examples of spheres and cones and triangles 
and squares. But the propositions do not 
merely apply to the actual figures printed in 
the book, but to any such figures. 

Thus geometry, like algebra, is dominated 
by the ideas of " any " and " some " things. 
Also, in the same way it studies the inter- 
relations of sets of things. For example, con- 
sider any two triangles ABC and DEF. 

236 



GEOMETRY 237 

What relations must exist between some of 
the parts of these triangles, in order that the 
triangles may be in all respects equal ? This 
is one of the first investigations undertaken 
in all elementary geometries. It is a study 





a c E f 

Fig. 33. 

of a certain set of possible correlations be- 
tween the two triangles. The answer is that 
the triangles are in all respects equal, if : 
Either, (a) Two sides of the one and the in- 
cluded angle are respectively equal to two 
sides of the other and the included angle : 

Or, (b) Two angles of the one and the side 
joining them are respectively equal to two 
angles of the other and the side joining them : 

Or, (c) Three sides of the one are respect- 
tively equal to three sides of the other. 

This answer at once suggests a further en- 
quiry. What is the nature of the correlation 
between the triangles, when the three angles 
of the one are respectively equal to the three 
angles of the other ? This further investiga- 
tion leads us on to the whole theory of simi- 



238 INTRODUCTION TO MATHEMATICS 

larity (cf. Chapter XIII.), which is another 
type of correlation. 

Again, to take another example, consider 
the internal structure of the triangle ABC. 
Its sides and angles are inter-related the 
greater angle is opposite to the greater side, 
and the base angles of an isosceles triangle 
are equal. If we proceed to trigonometry 
this correlation receives a more exact deter- 
mination in the familiar shape 

sin A sin B sin C 



a? = &2_j_ C 2 _ 2bccosA, with two similar 
formulae. 

Also there is the still simpler correlation 
between the angles of the triangle, namely, 
that their sum is equal to two right angles ; 
and between the three sides, namely, that the 
sum of the lengths of any two is greater than 
the length of the third 

Thus the true method to study geometry is 
to think of interesting simple figures, such as 
the triangle, the parallelogram, and the circle, 
and to investigate the correlations between 
their various parts. The geometer has in his 
mind not a detached proposition, but a figure 
with its various parts mutually inter-depend- 
ent. Just as in algebra, he generalizes the 
triangle into the polygon, and the side into 



GEOMETRY 239 

the conic section. Or, pursuing a converse 
route, he classifies triangles according as they 
are equilateral, isosceles, or scalene, and 
polygons according to their number of sides, 
and conic sections according as they are hy- 
perbolas, ellipses, or parabolas. 

The preceding examples illustrate how the 
fundamental ideas of geometry are exactly 
the same as those of algebra ; except that 
algebra deals with numbers and geometry 
with lines, angles, areas, and other geo- 
metrical entities. This fundamental identity 
is one of the reasons why so many geometrical 
truths can be put into an algebraic dress. 
Thus if A, B, and C are the numbers of degrees 
respectively in the angles of the triangle ABC, 
the correlation between the angles is repre- 
sented by the equation 



and if a, b, c are the number of feet respectively 
in the three sides, the correlation between the 
sides is represented by a <(6+c, b <(c-f a, 
c </*+&. Also the trigonometrical formulae 
quoted above are other examples of the same 
fact. Thus the notion of the variable and 
the correlation of variables is just as essential 
in geometry as it is in algebra. 

But the parallelism between geometry and 
algebra can be pushed still further, owing to 
the fact that lengths, areas, volumes, and 



240 INTRODUCTION TO MATHEMATICS 

angles are all measurable ; so that, for exam- 
ple, the size of any length can be determined 
by the number (not necessarily integral) of 
times which it contains some arbitrarily known 
unit, and similarly for areas, volumes, and 
angles. The trigonometrical formulae, given 
above, are examples of this fact. But it re- 
ceives its crowning application in analytical 
geometry. This great subject is often mis- 
named as Analytical Conic Sections, thereby 
fixing attention on merely one of its sub- 
divisions. It is as though the great science 
of Anthropology were named the Study of 
Noses, owing to the fact that noses are a 
prominent part of the human body. 

Though the mathematical procedures in 
geometry and algebra are in essence identical 
and intertwined in their development, there 
is necessarily a fundamental distinction be- 
tween the properties of space and the proper- 
ties of number in fact all the essential differ- 
ence between space and number. The " spaci- 
ness " of space and the " numerosity " of 
number are essentially different things, and 
must be directly apprehended. None of the 
applications of algebra to geometry or of 
geometry to algebra go any step on the road 
to obliterate this vital distinction. 

One very marked difference between space 
and number is that the former seems to be so 
much less abstract and fundamental than the 



GEOMETRY 241 

latter. The number of the archangels can be 
counted just because they are things. When 
we once know that their names are Raphael, 
Gabriel, and Michael, and that these distinct 
names represent distinct beings, we know with- 
out further question that there are three of 
them. All the subtleties in the world about 
the nature of angelic existences cannot alter 
this fact, granting the premisses. 

But we are still quite in the dark as to their 
relation to space. Do they exist in space at 
all ? Perhaps it is equally nonsense to say 
that they are here, or there, or anywhere, or 
everywhere. Their existence may simply have 
no relation to localities in space. According- 
ly, while numbers must apply to all things, 
space need not do so. 

The perception of the locality of things 
would appear to accompany, or be involved 
in many, or all, of our sensations. It is in- 
dependent of any particular sensation in the 
sense that it accompanies many sensations. 
But it is a special peculiarity of the things 
which we apprehend by our sensations. The 
direct apprehension of what we mean by the 
positions of things in respect to each other 
is a thing sui generis, just as are the appre- 
hensions of sounds, colours, tastes, and smells. 
At first sight therefore it would appear that 
mathematics, in so far as it includes geometry 
in its scope, is not abstract in the sense in 



242 INTRODUCTION TO MATHEMATICS 

which abstractness is ascribed to it in 
Chapter I. 

This, however, is a mistake ; the truth being 
that the " spaciness " of space does not enter 
into our geometrical reasoning at all. It 
enters into the geometrical intuitions of 
mathematicians in ways personal and peculiar 
to each individual. But what enter into the 
reasoning are merely certain properties of 
things in space, or of things forming space, 
which properties are completely abstract in 
the sense in which abstract was denned in 
Chapter I.; these properties do not involve 
any peculiar space-apprehension or space- 
intuition or space-sensation. They are on 
exactly the same basis as the mathematical 
properties of number. Thus the space-intui- 
tion which is so essential an aid to the study 
of geometry is logically irrelevant : it does 
not enter into the premisses when they are 
properly stated, nor into any step of the rea- 
soning. It has the practical importance of an 
example, which is essential for the stimulation 
of our thoughts. Examples are equally neces- 
sary to stimulate our thoughts on number. 
When we think of " two " and " three " we 
see strokes in a row, or balls in a heap, or 
some other physical aggregation of particular 
things. The peculiarity of geometry is the 
fixity and overwhelming importance of the 
one particular example which occurs to our 



GEOMETRY 243 

minds. The abstract logical form of the 
propositions when fully stated is, " If any 
collections of things have such and such 
abstract properties, they also have such and 
such other abstract properties.'-' But what 
appears before the mind's eye is a collection 
of points, lines, surfaces, and volumes in the 
space : this example inevitably appears, and 
is the sole example which lends to the propo- 
sition its interest. However, for all its over- 
whelming importance, it is but an example. 

Geometry, viewed as a mathematical science, 
is a division of the more general science of 
order. It may be called the science of dimen- 
sional order ; the qualification " dimensional " 
has been introduced because the limitations, 
which reduce it to only a part of the general 
science of order, are such as to produce the 
regular relations of straight lines to planes, 
and of planes to the whole of space. 

It is easy to understand the practical im- 
portance of space in the formation of the 
scientific conception of an external physical 
world. On the one hand our space-percep- 
tions are intertwined in our various sensations 
and connect them together. We normally 
judge that we touch an object in the same 
place as we see it ; and even in abnormal 
cases we touch it in the same space as we see 
it, and this is the real fundamental fact which 
ties together our various sensations. Accord- 



244 INTRODUCTION TO MATHEMATICS 

ingly, the space perceptions are in a sense the 
common part of our sensations. Again it 
happens that the abstract properties of space 
form a large part of whatever is of spatial 
interest. It is not too much to say that to 
every property of space there corresponds an 
abstract mathematical statement. To take 
the most unfavourable instance, a curve may 
have a special beauty of shape : but to this 
shape there will correspond some abstract 
mathematical properties which go with this 
shape and no others. 

Thus to sum up : (1) the properties of space 
which are investigated in geometry, like those 
of number, are properties belonging to things 
as things, and without special reference to 
any particular mode of apprehension : (2) 
Space-perception accompanies our sensations, 
perhaps all of them, certainly many ; but it 
does not seem to be a necessary quality of 
things that they should all exist in one space 
or in any space. 



CHAPTER XVII 

QUANTITY 

IN the previous chapter we pointed out 
that lengths are measurable in terms of some 
unit length, areas in term of a unit area, and 
volumes in terms of a unit volume. 

When we have a set of things such as 
lengths which are measurable in terms of any 
one of them, we say that they are quantities 
of the same kind. Thus lengths are quantities 
of the same kind, so are areas, and so are 
volumes. But an area is not a quantity of 
the same kind as a length, nor is it of the 
same kind as a volume. Let us think a little 
more on what is meant by being measurable, 
taking lengths as an example. 

Lengths are measured by the foot-rule. By 
transporting the foot-rule from place to place 
we judge of the equality of lengths. Again, 
three adjacent lengths, each of one foot, form 
one whole length of three feet. Thus to 
measure lengths we have to determine the 
equality of lengths and the addition of lengths. 
When some test has been applied, such as the 
transporting of a foot-rule, we say that the 
lengths are equal ; and when some process 

245 



has been applied, so as to secure lengths being 
contiguous and not overlapping, we say that 
the lengths have been added to form one 
whole length. But we cannot arbitrarily take 
any test as the test of equality and any 
process as the process of addition. The re- 
sults of operations of addition and of judg- 
ments of equality must be in accordance with 
certain preconceived conditions. For exam- 
ple, the addition of two greater lengths must 
yield a length greater than that yielded by 
the addition of two smaller lengths. These 
preconceived conditions when accurately for- 
mulated may be called axioms of quantity. 
The only question as to their truth or falsehood 
which can arise is whether, when the axioms 
are satisfied, we necessarily get what ordinary 
people call quantities. If we do not, then 
the name " axioms of quantity " is ill-judged 
that is all. 

These axioms of quantity are entirely ab- 
stract, just as are the mathematical properties 
of space. They are the same for all quantities, 
and they presuppose no special mode of per- 
ception. The ideas associated with the notion 
of quantity are the means by which a con- 
tinuum like a line, an area, or a volume can 
be split up into definite parts. Then these 
parts are counted ; so that numbers can be 
used to determine the exact properties of a 
continuous whole. 



QUANTITY 247 

Our perception of the flow of time and of 
the succession of events is a chief example 
of the application of these ideas of quantity. 
We measure time (as has been said in con- 
sidering periodicity) by the repetition of 
similar events the burning of successive 
inches of a uniform candle, the rotation of 
the earth relatively to the fixed stars, the 
rotation of the hands of a clock are all ex- 
amples of such repetitions. Events of these 
types take the place of the foot-rule in rela- 
tion to lengths. It is not necessary to assume 
that events of any one of these types are 
exactly equal in duration at each recurrence. 
What is necessary is that a rule should be 
known which will enable us to express the 
relative durations of, say, two examples of 
some type. For example, we may if we like 
suppose that the rate of the earth's rotation 
is decreasing, so that each day is longer than 
the preceding by some minute fraction of a 
second. Such a rule enables us to compare 
the length of any day with that of any other 
day. But what is essential is that one series 
of repetitions, such as successive days, should 
be taken as the standard series ; and, if the 
various events of that series are not taken as 
of equal duration, that a rule should be 
stated which regulates the duration to be 
assigned to each day in terms of the duration 
of any other day. 



248 INTRODUCTION TO MATHEMATICS 

What then are the requisites which such 
a rule ought to have ? In the first place it 
should lead to the assignment of nearly equal 
durations to events which common sense 
judges to possess equal durations. A rule 
which made days of violently different lengths, 
and which made the speeds of apparently 
similar operations vary utterly out of pro- 
portion to the apparent minuteness of their 
differences, would never do. Hence the first 
requisite is general agreement with common 
sense. But this is not sufficient absolutely 
to determine the rule, for common sense is a 
rough observer and very easily satisfied. The 
next requisite is that minute adjustments of 
the rule should be so made as- to allow of the 
simplest possible statements of the laws of 
nature. For example, astronomers tell us 
that the earth's rotation is slowing down, so 
that each day gains in length by some incon- 
ceivably minute fraction of a second. Their 
only reason for their assertion (as stated more 
fully in the discussion of periodicity) is that 
without it they would have to abandon the 
Newtonian laws of motion. In order to keep 
the laws of motion simple, they alter the 
measure of time. This is a perfectly legiti- 
mate procedure so long as it is thoroughly 
understood. 

What has been said above about the ab- 
stract nature of the mathematical properties 



QUANTITY 

of space applies with appropriate verbal 
changes to the mathematical properties of 
time. A sense of the flux of time accompanies 
all our sensations and perceptions, and prac- 
tically all that interests us in regard to time 
can be paralleled by the abstract mathe- 
matical properties which we ascribe to it. 
Conversely what has been said about the two 
requisites for the rule by which we determine 
the length of the day, also applies to the rule 
for determining the length of a yard measure 
namely, the yard measure appears to retain 
the same length as it moves about. Accord- 
ingly, any rule must bring out that, apart 
from minute changes, it does remain of in- 
variable length; Again, the second requisite 
is this, a definite rule for minute changes 
shall be stated which allows of the simplest 
expression of the laws of nature. For ex- 
ample, in accordance with the second re- 
quisite the yard measures are supposed to 
expand and contract with changes of tem- 
perature according to the substances which 
they are made of. 

Apart from the facts that our sensations 
are accompanied with perceptions of locality 
and of duration, and that lines, areas, volumes, 
and durations, are each in their way quanti- 
ties, the theory of numbers would be of very 
subordinate use in the exploration of the laws 
of the Universe,- As it is, physical science 



reposes on the main ideas of number, quan- 
tity, space, and time. The mathematical 
sciences associated with them do not form 
the whole of mathematics, but they are the 
substratum of mathematical physics as at 
present existing. 



NOTES 

A (p. 60). In reading these equations it must be noted 
that a bracket is used in mathematical symbolism to 
mean that the operations within it are to be performed 
first. Thus (l + 3)+2 directs us first to add 3 to 1, and 
then to add 2 to the result; and l + (3+2) directs us 
first to add 2 to 3, and then to add the result to 1. Again 
a numerical example of equation (5) is 

2x(3+4)=(2x3) + (2x4). 
We perform first the operations in brackets and obtain 

2x7=6+8 
which is obviously true. 

SP 
B (p. 136). This fundamental ratio -p^ is called the 

eccentricity of the curve. The shape of the curve, as 
distinct from its scale or size, depends upon the value of 
its eccentricity. Thus it is wrong to think of ellipses 
in general or of hyperbolas in general as having in either 
case one definite shape. Ellipses with different eccen- 
tricities have different shapes, and their sizes depend 
upon the lengths of their major axes. An ellipse with 
small eccentricity is very nearly a circle, and an ellipse 
of eccentricity only slightly less than unity is a long 
flat oval. All parabolas have the same eccentricity and 
are therefore of the same shape, though they can be 
drawn to different scales. 



BIBLIOGRAPHY 251 

C (p. 204). If a series with all its terms positive is 
convergent, the modified series found by making some 
terms positive and some negative according to any 
definite rule is also convergent. Each one of the set of 
series thus found, including the original series, is called 
" absolutely convergent." But it is possible for a series 
with terms partly positive and partly negative to be 
convergent, although the corresponding series with all 
its terms positive is divergent. For example, the series 

1-i+i-i+etc. 
is convergent though we have just proved that 

l+i-H+i+ etc. 

is divergent. Such convergent series, which are not 
absolutely convergent, are much more difficult to deal 
with than absolutely convergent series. 



BIBLIOGRAPHY 

NOTE ON THE STUDY OF MATHEMATICS 

THE difficulty that beginners find in the study of this 
science is due to the large amount of technical detail which 
has been allowed to accumulate in the elementary text- 
books, obscuring the important ideas. 

The first subjects of study, apart from a knowledge of 
arithmetic which is presupposed, must be elementary 
geometry and elementary algebra. The courses in both 
subjects should be short, giving only the necessary ideas ; 
the algebra should be studied graphically, so that in 
practice the ideas of elementary coordinate geometry are 
also being assimilated. The next pair of subjects should 
be elementary trigonometry and the coordinate geometry 
of the straight line and circle. The latter subject is a 
short one ; for it really merges into the algebra. The 
student is then prepared to enter upon conic sections, a 
very short course of geometrical conic sections and a longer 
one of analytical conies. But in all these courses great 
care should be taken not to overload the mind with more 



252 BIBLIOGRAPHY 

detail than is necessary for the exemplification of the 
fundamental ideas. 

The differential calculus and afterwards the integral 
calculus now remain to be attacked on the same system. 
A good teacher will already have illustrated them by the 
consideration of special cases in the course on algebra 
and coordinate geometry. Some short book on three- 
dimensional geometry must be also read. 

This elementary course of mathematics is sufficient for 
some types of professional career. It is also the necessary 
preliminary for any one wishing to study the subject for 
its intrinsic interest. He is now prepared to commence 
on a more extended course. He must not, however, hope 
to be able to master it as a whole. The science has grown 
to such vast proportions that probably no living mathe- 
matician can claim to have achieved this. 

Passing to the serious treatises on the subject to be read 
after this preliminary course, the following may be men- 
tioned : Cremona's Pure Geometry (English Translation, 
Clarendon Press, Oxford), Hobson's Treatise on Trigono- 
metry, ChrystaPs Treatise on Algebra (2 volumes), Salmon's 
Gonic Sections, Lamb's Differential Calculus, and some 
book on Differential Equations. The student will probably 
not desire to direct equal attention to all these subjects, 
but will study one or more of them, according as his interest 
dictates. He will then be prepared to select more ad- 
vanced works for himself, and to plunge into the higher 
parts of the subject. If his interest lies in analysis, he 
should now master an elementary treatise on the theory 
of Functions of the Complex Variable ; if he prefers to 
specialize in Geometry, he must now proceed to the 
standard treatises on the Analytical Geometry of three 
dimensions. But at this stage of his career in learning 
he will not require the advice of this note. 

I have deliberately refrained from mentioning any 
elementary works. They are very numerous, and of 
various merits, but none of such outstanding superiority 
as to require special mention by name to the exclusion 
of all the others. 



INDEX 



Abel, 156 
Abscissa, 95 

Absolute Convergence, 251 
Abstract Nature of Geo- 
metry, 242 et aeqq. 
Abstractness (defined), 9, 13 
Adams, 220 

Addition Theorem, 212 
Ahmes, 71 
Alexander the Great, 128, 

129 
Algebra.Fundamental Laws 

of, 60 
Ampere, 34 
3 Analytical Conic Sections, 

240 
Apollonius of Perga, 131, 

134 

j Approximation, 197 et aeqq. 
Arabic Notation, 58 et aeqq. 
Archimedes, 37 et aeqq. 
, Argument of a Function, 

146 

i Aristotle, 30, 42, 128 
Astronomy, 137, 173, 174 
Axes, 125 
Axioms of Quantity, 246 

et aeqq. 

Bacon, 156 
Ball, W. W. R., 58 
Beaconsfield, Lord, 41 
Berkeley, Bishop, 226 
Bhaskara, 58 



Cantor, Georg, 79 

Circle, 120, 130, 180 et 

aeqq. 

Circular Cylinder, 143 
Clerk Maxwell, 34, 35 
Columbus, 122 
Compact Series, 76 
Complex Quantities, 109 
.Conic Sections, 128 et aeqq. 
Constants, 69, 117 
Continuous Functions, 150 

et aeqq. ; 162 (defined) 
Convergence, Absolute, 251 
Convergent, 203 et aeqq. 
Coordinate Geometry, 112 

et aeqq. 

Coordinates, 95 
Copernicus, 45, 137 
Coaine, 182 et aeqq. 
Coulomb, 33 
Cross Ratio, 140 

Darwin, 138, 220 
Derived Function, 234 
Descartes, 95, 113, 116, 

122, 218 
Differential Calculus, 217 

et aeqq. 

Differential Coefficient, 234 
Directrix, 135 
Discontinuous Functions, 

150 et aeqq. 
Distance, 30 
Divergent, 203 et aeqq. 



253 



254 



INDEX 



Dynamical Explanation, 

13, 14, 47 et segq. 
Dynamics, 30, 43 et seqq. 

Eccentricity, 250 
Electric Current, 33 
Electricity, 32 et seqq. 
Electromagnetism, 31 et 

seqq. 
Ellipse, 45, 120, 130 et 

seqq. 

Euclid, 114 
Exponential Series, 211 

et seqq. 

Faraday, 34 

Format, 218 

Fluxions, 219 

Focus, 120, 135 

Force, 30 

Form, Algebraic, 66 et seqq., 

82, 117 

Fourier's Theorem, 191 
Fractions, 71 et seqq. 
Franklin, 32, 122 
Function, 144 et seqq. 

Galileo, 30, 42 et seqq., 122 

Galvani, 33 

Generality in Mathematics, 

82 
Geometrical Series, 206 

et seqq. 

Geometry, 36, 236 et seqq. 
Gilbert, Dr., 32 
Graphs, 148 et seqq. 
Gravitation, 29, 139 



Halley, 139 

Harmonic Analysis, 192 

Harriot, Thomas, 66 



Here, 35 
Hiero, 38 
Hipparchus, 173 
Hyperbola, 131 et seqq. 

Imaginary Numbers, 87 et 

seqq. 

Imaginary Quantities, 109 
Incommensurable Ratios, 

72 et seqq. 
Infinitely Small Quantities, 

226 et seqq. 

Integral Calculus, 222 
Interval, 158 et seqq. 



Kepler, 45, 46, 137, 138 
Kepler's Laws, 138 



Laputa, 10 

Laws of Motion, 167 et seqq., 

248 

Leibniz, 16, 218 et seqq. 
Leonardo da Vinci, 42 
Leverrier, 220 
Light, 35 
Limit of a Function, 227 

et seqq. 
Limit of a Series, 199 

et seqq. 
Limits, 77 
Locus, 121 etseqq., 141 



Macaulay, 156 
Malthus, 138 
Marcellus, 37 
Mass, 30 
Mechanics, 46 
Menaechmus, 128, 129 
Motion, First Law of, 43 



INDEX 



255 



Neighbourhood, 159 et aeqq. 
Newton, 10, 16, 30, 34, 37, 

38, 43, 46, 139, 218 et aeqq. 
Non-Uniform Convergence, 

208 et aeqq. 
Normal Error, Curva of, 

214 

Oersted, 34 

Order, 194 et aeqq. 

Order, Type of, 75 et aeqq., 

196 

Ordered Couples, 93 et aeqq. 
Ordinate, 95 
Origin, 95, 126 

Pappus, 135, 136 
Parabola, 131 et aeqq. 
Parallelogram Law, 61 et 

aeqq., 99, 126 
Parameters, 69, 117 
Pencils, 140 
Period, 170, 189 etaeqq. 
Periodicity, 164 et seqq., 

188, 216 

Pitt, William, 194 
Pizarro, 122 
Plutarch, 37 
Positive and Negative 

Numbers, 83 et aeqq. 
Projective Geometry, 139 
Ptolemy, 137, 173 
Pythagoras, 18 

Quantity, 245 et aeqq. 

Rate of Increase of Func- 
tions, 220 et aeqq. 
Ratio, 72 et aeqq. 
Real Numbers, 73 et aeqq. 
Rectangle, 57 



Relations between Vari- 
ables, 18 et aeqq. 
Resonance, 170, 171 
Rosebery, Lord, 194 

Scale of a Map, 178 

Seidel, 210 

Series, 74 et aeqq., 194 et 
aeqq. 

Shelley (quotation from), 
217 

Similarity, 177 et seqq., 
237 

Sine, 182 et aeqq. 

Specific Gravity, 41 

Squaring the Circle, 187 

Standard of Approxima- 
tion, 159 et aeqq., 201 et 
aeqq., 229 etseqq. 

Steps, 79 et aeqq., 96 

Stifel, 85 

Stokes, Sir George, 210 

Sum to Infinity, 201 et 
aeqq. 

Surveys, 176 et aeqq. 

Swift, 10 

Tangents, 221, 222 
Taylor's Theorem, 166, 

157 
Time, 166 et aeqq., 247 

et aeqq. 
Transportation, Vector of, 

54 et aeqq. 

Triangle, 176 et aeqq., 237 
Triangulation, 177 
Trigonometry, 173 et aeqq. 

Uniform Convergence, 208 

et aeqq. 
Unknown, The, 17, 23 



256 



INDEX 



Value of a Function, 146 
Variable, The, 18, 24, 49, 

82, 234, 239 
Variable Function, 147 
Vectors, 51 et eeqq., 85, 96 
Vertex, 134 



Volta, 33 

Wallace, 220 
Weierstrass, 156, 226, 228 

Zero, 63 et seqq., 108 



Printed fry Hazell, Walton & Viney, Ld., London and Ayktbwy. 



The 

Home University 
Library 



of Modern 
Knowledge 



jj Comprehensive Series of New 
and Specially Written (Books 



EDITORS : 

PROF. GILBERT MURRAY, D.Litt., LL.D., F.B.A. 
HERBERT FISHER, M.A., F.B.A. 
PKOF. J. ARTHUR THOMSON, M.A. 
PROF. WM. T. BREWSTER, M.A. 

The Home University Library 

" Is without the slightest doubt the pioneer in supplying serious literature 
for a large section of the public who are interested in the liberal educa- 
tion of the State." The Daily Mail. 

" It is a thing very favourable to the real success of The Home 
University Library that its volumes do not merely attempt to feed 
ignorance with knowledge. The authors noticeably realise that the 
simple willing appetite of sharp-set ignorance is not specially common 
nowadays ; what is far more common is a hunger which has been 
partially but injudiciously filled, with more or less serious results of 
indigestion. The food supplied is therefore frequently medicinal as 
well as nutritious ; and this is certainly what the time requires. " 
Manchester Guardian. 

"Each volume represents a three-hours' traffic with the talking-power 
of a good brain, operating with the ease and interesting freedom of a 
specialist dealing with his own subject. ... A series which promises to 
perform a real social service." The Times. 

"We can think of no series now being issued which better deserves 
support." The Observer. 



We think if they were given as prizes in place of the more costly 

di: 

id prol 
series they might well take : 



ispensed on prize days, the pupils would 
rofit. If the publishers want a motto for the 



rubbish that is wont to be 

find more pleasure and 

series they might well take : ' Infinite riches in a little room.'" Irish 

Journal of Education, 

" The scheme was successful at the start because it met a want 
among earnest readers; but its wider and* sustained success, surely, 
comes from the fact that it has to a large extent created and certainly 
refined the taste by which it is appreciated." Daily Chronicle. 

" Here is the world's learning in little, and none too poor t< 
house-room!" Daily Telegraph. 



to give it 



]/- net 
in cloth 


256 Pages 


2/6 net 
in leather 



History and (geography 



3. THE FRENCH REVOLUTION 

By HILAIKE BELLOC, M.A. (With Maps.) "It is coloured with all the 
militancy of the author's temperament." Daily News. 

4. HISTORY OF WAR AND PEACE 

By G. H. FERRIS. The Rt. Hon. JAMES BRYCE writes: " I have read it with 
much interest and pleasure, admiring the skill with which you have managed 
to compress so many facts and views into so small a volume." 

8. POLAR EXPLORATION 

By Dr W. S. BRUCE, F.R.S.E., Leader of the "Scotia" Expedition. (With 
Maps.) "A very freshly written and interesting narrative." The Times. 
"A fascinating book." Portsmouth Times. 

12. THE OPENING-UP OF AFRICA 

By Sir H. H. JOHNSTON. G.C.M.G., K.C.B., D.Sc., F.Z.S. (With Maps.) 
" The Home University Library is much enriched by this excellent work." 
Daily Mail. 

13. MEDIAEVAL EUROPE 

By H. W. C. DAVIS, M.A. (With Maps.) "One more illustration of the 
fact that it takes a complete master of the subject to write briefly upon it." 
Manchester Guardian. 

14. THE PAPACY &* MODERN TIMES (1303-1870) 

By WILLIAM BARRY, D.D. "Dr Barry has a wide range of knowledge 
and an artist's power of selection." Manchester Guardian. 

23. HISTORY OF OUR TIME, 1885-1911 

By G. P. GOOCH, M.A. " Mr Gooch contrives to breathe vitality into his story, 
and to give us the flesh as well as the bones of recent happenings." Observer. 

25. THE CIVILISATION OF CHINA 

By H. A. GILES, LL.D., Professor of Chinese in the University of Cambridge. 
"In all the mass of facts, Professor Giles never becomes dull. He is always 
ready with a ghost story or a street adventure for the reader's recreation." 
Spectator. 

29. THE DA WN OF HISTORY 

By J.L.MYRES, M. A., F.S. A., Wykeham Professor of Ancient History, Oxford. 
"There is not a page in it that is not suggestive." Manchester Guardian. 

33. THE HISTORY OF ENGLAND: 
A Study in Political Evolution. 

By Prof. A. F. POLLARD, M.A. With a Chronological Table. " It takes its 
place at once among the authoritative works on English history." Observer. 

34. CANADA 

By A. G. BRADLEY. " Who knows Canada, better than Mr A. G. Bradley? " 
Daily Chronicle. "The volume makes an immediate appeal to the man who 
wants to know something vivid and true about Canada." Canadian Gazette. 



37. PEOPLES 6* PROBLEMS OF INDIA 

By Sir T. W. HOLDERNESS, K.C.S.I., Secretary of the Revenue, Statistics, 
! and Commerce Department of the India Office. "Just the book which news- 
paper readers require to-day, and a marvel of comprehensiveness." Pall 
\ Mall Gazette. 

42. ROME 

By W. WARDE FOWLHR, M.A. " A masterly sketch of Roman character and 
of what it did for the world." The Spectator. "It has all the lucidity and 
charm of presentation we expect from this writer." Manchester Guardian. 

48. THE AMERICAN CIVIL WAR 

By F. L. PAXSON, Professor of American History, Wisconsin University. 
(With Maps.) "A stirring study." The Guardian. 

51. WARFARE IN BRITAIN 

By HILAIRE BELLOC, M.A. An account of how and where great battles of the 
past were fought on British soil, the roads and physical conditions determining 
the island's strategy, the castles, walled towns, etc. 

55. MASTER MARINERS 

By J. R. SPEARS. The romance of the sea, the great voyages of discovery, 
naval battles, the heroism of the sailor, and the development of the ship, from 
ancient times to to-day. 

IN PREPARATION 

ANCIENT GREECE. By Prof. GILBERT MURRAY, D.Litt., LL.D., F.B.A 
ANCIENT EGYPT. By F. LL. GRIFFITH, M.A. 
THE ANCIENT EAST. By D. G. HOGARTH, M.A., F.B.A. 
A SHORT h'ISTOR YOFEUROPE. By HERBERT FISHER, M. A., F.B.A. 
PREHISTORIC BRITAIN. By ROBERT MUNRO, M.A., M.D., LL.D. 
THE BYZANTINE EMPIRE. By NORMAN H. BAVNES. 
THE REFORM A TION. By Principal LINDSAY, LL.D. 
NAPOLEON. By HERBERT FISHER, M.A., F.B.A. 
A SHORT HISTORY OF RUSSIA. By Prof. MILYOUKOV. 
MODERN TURKEY. By D. G. HOGARTH, M.A. 
FRANCE OF TO-DAY. By ALBERT THOMAS. 
GERMANY OF TO-DA Y. By CHARLES TOWER. 
THE NAVY AND SEA POWER. By DAVID HANNAY. 
HISTORY OF SCOTLAND. By R. S. RAIT, M.A. 
SOUTH AMERICA. By Prof. W. R. SHEPHERD. 
LONDON. By Sir LAURENCE GOMME, F.S.A. 

HISTORY AND LITERATURE OF SPAIN. By J. FITZMAURICE- 
KELLY, F.B.A., Litt.D. 



Literature and 



2. SHAKESPEARE 

By JOHN MASEFIELD. " The book is a joy. We have had half-a-dozen more 
learned books on Shakespeare in the last few years, but not one so wise." 
Manchester Guardian. 

27. ENGLISH LITERATURE: MODERN 

By G. H. MAIR, M.A. " Altogether a fresh and individual book." Olstrver. 

35. LANDMARKS IN FRENCH LITERATURE 

By G. L. STRACHEY. " Mr Strachey is to be congratulated on his courage and 
success. It is difficult to imagine how a better account of French Literature 
could be given in 250 small pages than he has given here." The Times. 



39- ARCHITECTURE 

By Prof. W. R. LETHABY. (Over forty Illustrations.) " Popular guide-books 
to architecture are, as a rule, not worth ranch. This volume is a welcome excep- 
tion." Building News. " Delightfully bright reading." Christian World. 

43. ENGLISH LITERATURE: MEDIAEVAL 

By Prof. W. P. KER, M.A. "Prof. Ker has long proved his worth as one of 
the soundest scholars in English we have, and he is the very man to put an 
outline of English Mediaeval Literature before the uninstructed public. His 
knowledge and taste are unimpeachable, and his style is effective, simple, yet 
never dry." The Athemeum. 

45. THE ENGLISH LANGUAGE 

By L. PEARSALL SMI-TH, M.A. "A wholly fascinating study of the different 
streams that went to the making of the great river of the English speech." 
Daily News. 

52. GREAT WRITERS OF AMERICA 

By Prof. J. EKSKINE and Prof. W. P. TRENT. A popular sketch by two 
foremost authorities. 

IN PREPARATION 

ANCIENT ART AND RITUAL. By Miss JANE HARRISON, LL.D., 

D.Litt. 

GREEK LITERA TURE. By Prof. GILBERT MURRAY, D.Litt. 
LA TIN LITER A TURE. By Prof. J. S. PHILLIMORE. 
CHA UCER AND HIS TIME. By Miss G. E. HADOW. 
THE RENAISSANCE. By Mrs R. A. TAYLOR. 

ITALIAN A RTOF THE RENAISSANCE. By ROGER E. FRY, M.A. 
THE ART OF PAINTING. By Sir FREUERICK WEDMORE. 
DR JOHNSON AND HIS CIRCLE. By JOHN BAILEY, M.A. 
THE VIC IORIAN AGE. By G. K- CHESTERTON. 
ENGLISH COMPOSITION. By Prof. WM. T. BREWSTER. 
GREA T WRITERS OF RUSSIA. By C. T. HAGBERG WRIGHT, LL.D. 
THE LITERATURE OF GERMANY. By Prof. J. G. ROBERTSON, 

M.A., Ph.D. 
SCANDINAVIAN HISTORY AND LITERATURE. By T. C. 

SNOW, M.A. 



Science 



7. MODERN GEOGRAPHY 

By Dr MARION NEWBIGIN. (Illustrated.) "Geography, again : what a dull, 
tedious study that was wont to be I . . . But Miss Marion Newbigin invests its 
dry bones with the flesh and blood of romantic interest, taking stock of 
geography as a fairy-book of science." Daily Telegraph. 

9. THE EVOLUTION OF PLANTS 

By Dr D. H. SCOTT, M.A., F.R.S., late Hon. Keeper of the Jodrell Laboratory, 
Kew. (Fully illustrated.) "The information which the book provides is as 
trustworthy as first-band knowledge can make it. ... Dr Scott's candid and 
familiar style makes the difficult subject both fascinating and easy." 
Gardeners' Chronicle. 

17. HEALTH AND DISEASE 

By W. LESLIE MACKKNZIE, M.D., Local Government Board, Edinburgh. 
"The science of public health administration has had no abler or more attractive 
exponent than Dr Mackenzie. He adds to a thorough grasp of the problems 
an illuminating style, and an arresting manner of treating a subject often 
dull and sometimes unsavoury." Economist. 



1 8. INTRODUCTION TO MATHEMATICS 

' By A. N. WHITEHEAD, Sc.D., F.R.S. (With Diagrams.) "MrWhitehead 
has discharged with conspicuous success the task he is so exceptionally qualified 

I to undertake. For he is one of our great authorities upon the foundations of the 
science, and has the breadth of view which is so requisite in presenting to the 
reader its aims. His exposition is clear and striking." Westminster Gazette. 

19. THE ANIMAL WORLD 

By Professor F. W. GAMBLE, D.Sc., F.R.S. With Introduction hy Sir Oliver 
Lodge. (Many Illustrations.) " A delightful and instructive epitome of animal 
(and vegetable) life. ... A most fascinating and suggestive survey." Morning 
Post. 

20. EVOLUTION 

By Professor J. ARTHUR THOMSON and Professor PATRICK GEDDES. "A 
many-coloured and romantic panorama, opening up, like no other book we know, 
a rational vision of world-development." Belfast News-Letter. 

22. CRIME AND INSANITY 

By Dr C. A. MERCIER, F.R.C.P., F.R.C.S., Author of "Text-Book of In- 
sanity," etc- " Furnishes much valuable information from one occupying the 
highest position among medico-legal psychologists." Asylum NCVJS. 

28. PSYCHICAL RESEARCH 



and thus what he has to say on thought-reading, hypnotism, telepathy, crystal- 
vision, spiritualism, divinings, and so on, will be read with avidity." Dundee 




31. ASTRONOMY 

By A. R. HINKS, M.A., Chief Assistant, Cambridge Observatory. "Original 
in thought, eclectic in substance, and critical in treatment. . . . No better 
little book is available." School World. 

32. INTRODUCTION TO SCIENCE 

By J. ARTHUR THOMSON, M.A., Regius Professor of Natural History, Aberdeen 
University. " Professor Thomson's delightful literary style is well known ; and 
here he discourses freshly and easily on the methods of science and its relations 
with philosophy, art, religion, and practical life." Aberdeen Journal, 

36. 



By H. N. DICKSON, D.Sc. Oxon., M.A., F.R.S.E., President of the Royal 
Meteorological Society ; Professor of Geography in University College, Reading. 
(With Diagrams.) "The author has succeeded in presenting in a very lucid 
and agreeable manner the causes of the movement of the atmosphere and of 
the more stable winds." Manchester Guardian. 

41. ANTHROPOLOGY 

By R R. MARETT, M.A., Reade 
"An absolutely perfect handboo 
fascinating and human that it bea 

44. THE PRINCIPLES OF PHYSIOLOGY 

By Prof. J. G. McKENDRiCK, M.D. " It is a delightful and wonderfully com- 
prehensive handling of a subject which, while of importance to all, does not 
readily lend itself to untechnical explanation. . . . The little book is more than 
a mere repository of knowledge ; upon every page of it is stamped the impress 
of a creative imagination." Glasgow Herald. 



By R. R. MARETT, M.A., Reader in Social Anthropology in Oxford University. 
"An absolutely perfect handbook, so clear that a child could understand it, so 
fascinating and human that it beats fiction ' to a frazzle.' " Morning Leader. 



46. MATTER AND ENERGY 

By F. SODDY, M.A., F.R.S. "A most fascinating and instructive account or 
the great facts of physical science, concerning which our knowledge, of later 
years, has made such wonderful progress." The Bookseller. 

49. PSYCHOLOGY, THE STUDY OF BEHAVIOUR 

By Prof. W. McDouGALL, F.R.S., M.B. "A happy example of the non- 
technical handling of an unwieldy science, suggesting rather than dogmatising. 
It should whet appetites for deeper study." Christian World. 

53. THE MAKING OF THE EARTH 

ByProf.J.W. GREGORY, F.R.S. (With 38 Maps and Figures.) The Professor 
of Geology at Glasgow describes the origin of the earth, the formation and 
changes of its surface and structure, its geological history, the first appearance 
of life, and its influence upon the globe. 

57. THE HUMAN BODY 

By A. KEITH, M.D., LL,D., Conservator of Museum and Hunterian Pro- 
fessor, Royal College of Surgeons. (Illustrated.) The work of the dissecting- 
room is described, and among other subjects dealt with are : the development 
of the body ; malformations and monstrosities ; changes of youth and age ; sex 
differences, are they increasing or decreasing? race characters ; bodily features 
as indexes of mental character ; degeneration and regeneration ; and the 
genealogy and antiquity of man. 

58. ELECTRICITY 

By GisBERT KAPP, D.Eng., M.I.E.E., M.I.C.E., Professor of Electrical 
Engineering in the University of Birmingham. (Illustrated.) Deals with 
frictional and contact electricity ; potential ; electrification by mechanical 
means ; the electric current ; the dynamics of electric currents ; alternating 
currents ; the distribution of electricity, etc. 

IN PREPARATION 

CHEMISTRY. Py Prof. R. MELDOLA, F.R.S. 

THE MINERAL WORLD. By Sir T. H. HOLLAND, K.C.I. E., D.Sc. 

PLANT LII-'E. By Prof. J. B. FARMER, F.R.S. 

NERVES. By Prof. D. FRASER HARRIS, M.D., D.Sc. 

A STUDY OF SEX. By Prof. J. A. THOMSON and Prof. PATRICK GEDDES. 

THE GROWTH OF EUROPE. By Prof. GRKNVILLE COLE. 



Philosophy and "Religion 



ig's 
:tate 



15. MOHAMMEDANISM 

By Prof. D. S. MARGOLIOUTH, M.A., D.Litt. "This generous shilling': 
worth of wisdom. ... A delicate, humorous, and most responsible tractati 
by an illuminative professor." Daily Mail. 

40. THE PROBLEMS OF PHILOSOPHY 

By the Hon. BERTRAND RUSSELL, F.R.S. : 'A book that the ' man in the 
street ' will recognise at once to be a boon. . . . Consistently lucid and non- 
technical throughout." Christian World. 

47. BUDDHISM 



go. NONCONFORMITY: Its ORIGIN and PROGRESS 

I'.'- Principal W. B. SELBIE, M.A. "The historical part is brilliant in its 
:., clarity, and proportion, and in the later chapters on the present position 
.urns of Nonconformity Dr Selbie proves himself to be an ideal exponent 
of sound and moderate views." Christian World. 

54. ETHICS 

By G. E. MOORE, M.A., Lecturer in Moral Science in Cambridge University. 
Discusses Utilitarianism, the Objectivity of Moral Judgments, the Test of 
Right and Wrong, Free Will, and Intrinsic Value. 

56. THE MAKING OF THE NEW TESTAMENT 

By Prof. B. W. BACON, LL. LX, D.D. An authoritative summary of the results 
of modern critical research with regard to the origins of the New Testament, in 
" the formative period when conscious inspiration was still in its full glow rather 
than the period of collection into an official canon," showing the mingling of the 
two great currents of Christian thought " Pauline and 'Apostolic,' the Greek- 
Christian gospel about Jesus, and the Jewish-Christian gospel of Jesus, the 
gospel of the Spirit and the gospel of au thority." 

jo. MISSIONS: THEIR RISE and DEVELOPMENT 

By Mrs CREIGHTON. The beginning of modern missions after the Reforma- 
tion and their growth are traced, and an account is given of their present 
work, its extent and character. 

IN PREPARATION 

THE OLD TESTAMENT. By Prof. GEORGE MOORE, D.D., LL.D. 
BETWEEN THE OLD AND NEW TESTAMENTS. By R. H. 

CHARLES, D.D. 

COMPARATIVE RELIGION. By Prof. J. ESTLIN CARPENTER, D.Litt. 
A HISTOR Y of FREEDOM of THOUGHT. By Prof. J. B. BURY, LL.D. 
A HISTORY OF PHILOSOPHY. By CLEMKNT WKBB, M.A. 



Social Science 



. PARLIAMENT 

Its History, Constitution, and Practice. By Sir COURTENAY P. ILBERT. 
K.C.B., K.C.S.I., Clerk of the House of Commons. "The best book on the 
history and practice of the House of Commons since Bagehot's 'Constitution.'" 
Yorkshire Post. 

. THE STOCK EXCHANGE 

By F. W. HIRST, Editor of " The Economist." " To an unfinancial mind must 
be a revelation. . . . The book is as clear, vigorous, and sane as Bagehot's ' Lom- 
bard Street,' than which there is no higher compliment." Morning Leader 

. IRISH NATIONALITY 

By Mrs J. R. GREEN. " As glowing as it is learned. No book could be more 
timely." Daily News. "A powerful study. . . . A magnificent demonstration 
of the deserved vitality of the Gaelic spirit." Freeman s Journal. 

3. THE SOCIALIST MOVEMENT 

RAMSAY MACDONALD, M.T. "Admirably adapted for the purpose of 
exposition." The Times. "Mr MacDonald is a very lucid exponent. . . . The 
volume will be of great use in dispelling illusions about the tendencies of 
Socialism in this country." The Nation. 

i. CONSERVATISM 

Jy Lord HUGH CECIL, M.A., M.P. "One of those great little books which 
seldom appear more than once in a generation." Morning Post. 



1 6. THE SCIENCE OF WEALTH 

By J. A. HOUSON, M.A. "Mr J. A. Hobson holds an unique position among 
living economists. . . . The text-book produced is altogether admirable. 
Original, reasonable, and illuminating." The Nation. 

21. LIBERALISM 

By L. T. HOBHOUSE, M. A., Professor of Sociology in the University of London. 
"A book of rare quality. . . . We have nothing but praise for the rapid and 
masterly summaries of the arguments from first principles which form a large 
part of this book." Westminster Gazette. 

24. THE EVOLUTION OF INDUSTRY 

ByD. H. MACGREGCR, M.A., Professor of Political Economy in the University 
of Leeds. "A volume so dispassionate in terms may be read with profit by all 
interested in the present state of unrest." Aberdeen Journal. 

26. AGRICULTURE 

By Prof. W. SOMERVILLE, F.L.S. " It makes the results of laboratory work 
at the University accessible to the practical farmer." Athena-urn. 

30. ELEMENTS OF ENGLISH LA W 

By W. M. GELDART, M.A., B.C.L., Vinerian Professor of English Law at 
Oxford. "Contains a very clear account of the elementary principles under- 
lying the rules of English law ; and we can recommend it to all who wish to 
become acquainted with these elementary principles with a minimum of 
trouble." Scots Law Times. 

38. THE SCHOOL 

An Introduction to the Study of Education. 

By J. J. FINDLAY, M.A., Ph.D., Professor of Education in Manchester 
University. <: An amazingly comprehensive volume. . . . It is a remarkable 
performance, distinguished in its crisp, striking phraseology as well as its 
inclusiveness of subject-matter." Morning Post. 

-59. ELEMENTS OF POLITICAL ECONOMY 

By S. J. CHAPMAN, M.A., Professor of Political Economy in Manchester 
University. A simple explanation, in the light of the latest economic thought, 
of the working of demand and supply ; the nature of monopoly ; money and 
international trade ; the relation of wages, profit, interest, and rent ; and the 
effects of labour combination prefaced by a short sketch of economic study 
since Adam Smith. 

IN PREPARATION 

THE CRIMINAL AND THE COMMUNITY. By Viscount ST. 

CYRES, M.A. 

COMMONSENSE IN LA W. By Prof. P. VINOGRADOFF, D.C.L. 
THE CIVIL SERVICE. By GRAHAM WALLAS, M.A. 
PRACTICAL IDEALISM. By MAURICE HEWLETT. 
NEWSPAPERS. By G. BINNEY DIBBLEE. 
ENGLISH VILLAGE LIFE. By E. N. BENNETT, M.A. 
CO -PARTNERSHIP At\D PROFIT-SHARING. By ANEURIN 

WILLIAMS, J.P. 

THE SOCIAL SETTLEMENT. By JANE ADDAMS and R. A. WOODS. 
GREA T INVENTIONS. By Prof. J. L. MYRES, M.A., F.S.A. 
TOWN PLANNING. By RAYMOND UNWIN. 
POLITICAL THOUGHT IN ENGLAND: From Bentham to J. S. 

Mill. By Prof. W. L. DAVIDSON. 
POLITICAL THOUGHT IN ENGLAKD: From Herbert Spencer 

to To-day. By ERNEST BARKER, M.A. 

London: WimTMS^AND~NORGATE 

And of all Bookshops and Bookstalls. 






THE LIBRARY 
UNIVERSITY OF CALIFORNIA 

Santa Barbara 



THIS BOOK IS DUE ON THE LAST DATE 

STAMPED BELOW. 




UCT91989 

SETOSE? 2 



3 1205 00394 1521 






UC SOUTHERN REGIONAL LIBRARY FACILITY 



A 000 804 550 2